3023:
2400:
2379:
10963:
1491:. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions with continuous domains. This framework eventually became modern
8763:
2432:
1989:
3823:
10519:
57:
7982:
7607:
10418:
being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition and to find the symbolic answer whenever it exists. The Risch algorithm, implemented in
3501:
2266:
10538:, replaces the rectangles used in a Riemann sum with trapezoids. The trapezoidal rule weights the first and last values by one half, then multiplies by the step width to obtain a better approximation. The idea behind the trapezoidal rule, that more accurate approximations to the function yield better approximations to the integral, can be carried further:
3818:{\displaystyle {\begin{alignedat}{3}&f^{+}(x)&&{}={}\max\{f(x),0\}&&{}={}{\begin{cases}f(x),&{\text{if }}f(x)>0,\\0,&{\text{otherwise,}}\end{cases}}\\&f^{-}(x)&&{}={}\max\{-f(x),0\}&&{}={}{\begin{cases}-f(x),&{\text{if }}f(x)<0,\\0,&{\text{otherwise.}}\end{cases}}\end{alignedat}}}
3031:
limit of the integrals of the approximations. However, many functions that can be obtained as limits are not
Riemann-integrable, and so such limit theorems do not hold with the Riemann integral. Therefore, it is of great importance to have a definition of the integral that allows a wider class of functions to be integrated.
6436:
7170:
3047:
I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the
Riemann integral. But I can proceed differently. After I have taken all the money out of
2007:
Integrals appear in many practical situations. For instance, from the length, width and depth of a swimming pool which is rectangular with a flat bottom, one can determine the volume of water it can contain, the area of its surface, and the length of its edge. But if it is oval with a rounded bottom,
10499:
is a generalization of
Ramanujan's master theorem that can be applied to a wide range of univariate and multivariate integrals. A set of rules are applied to the coefficients and exponential terms of the integrand's power series expansion to determine the integral. The method is closely related to
3030:
It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the
2103:
2271:
which is larger than the exact value. Alternatively, when replacing these subintervals by ones with the left end height of each piece, the approximation one gets is too low: with twelve such subintervals the approximated area is only 0.6203. However, when the number of pieces increases to infinity,
10417:
provides a general criterion to determine whether the antiderivative of an elementary function is elementary and to compute the integral if is elementary. However, functions with closed expressions of antiderivatives are the exception, and consequently, computerized algebra systems have no hope of
8542:
The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces inside higher-dimensional spaces. Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with
2444:
There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but are also occasionally for pedagogical reasons. The most commonly used definitions are
10494:
Rule-based integration systems facilitate integration. Rubi, a computer algebra system rule-based integrator, pattern matches an extensive system of symbolic integration rules to integrate a wide variety of integrands. This system uses over 6600 integration rules to compute integrals. The
6137:
7691:
A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration. An improper integral occurs when one or more of these conditions is not satisfied. In some cases such integrals may be defined by considering the
8535:
5809:
6208:
10949:
10559:
polynomial. This polynomial is chosen to interpolate the values of the function on the interval. Higher degree Newton–Cotes approximations can be more accurate, but they require more function evaluations, and they can suffer from numerical inaccuracy due to
4222:
2619:
9550:
Unlike the cross product, and the three-dimensional vector calculus, the wedge product and the calculus of differential forms makes sense in arbitrary dimension and on more general manifolds (curves, surfaces, and their higher-dimensional analogs). The
1547:). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the
9499:
in the sense that the wedge product of two forms representing oriented lengths represents an oriented area. A two-form can be integrated over an oriented surface, and the resulting integral is equivalent to the surface integral giving the flux of
4402:
1841:
relates the evaluation of definite integrals to indefinite integrals. There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article).
9662:
of a three-dimensional object that has a curved boundary. The area of a two-dimensional region can be calculated using the aforementioned definite integral. The volume of a three-dimensional object such as a disc or washer can be computed by
6953:
7946:
3034:
Such an integral is the
Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus
5947:
2261:{\displaystyle \textstyle {\sqrt {\frac {1}{5}}}\left({\frac {1}{5}}-0\right)+{\sqrt {\frac {2}{5}}}\left({\frac {2}{5}}-{\frac {1}{5}}\right)+\cdots +{\sqrt {\frac {5}{5}}}\left({\frac {5}{5}}-{\frac {4}{5}}\right)\approx 0.7497,}
7815:
10290:
Sometimes it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable. Techniques include
2995:
9398:
6942:
5629:
1978:
5610:
3490:
3107:, so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.
8514:
5473:
7685:
8798:. The value of the surface integral is the sum of the field at all points on the surface. This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums.
4588:
Linearity, together with some natural continuity properties and normalization for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of
5282:
4990:
8939:
5147:
9235:
are functions in three dimensions. A differential one-form can be integrated over an oriented path, and the resulting integral is just another way of writing a line integral. Here the basic differentials
6709:
1388:. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a
4851:
2339:
8757:
10133:
9545:
10374:
that are specifically designed to perform difficult or tedious tasks, including integration. Symbolic integration has been one of the motivations for the development of the first such systems, like
9218:
3285:
2724:
196:
1217:
370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. This method was further developed and employed by
9469:
8944:
The fluid flux in this example may be from a physical fluid such as water or air, or from electrical or magnetic flux. Thus surface integrals have applications in physics, particularly with the
6958:
8657:
10285:
9956:
7585:
3379:
9817:
4055:
2729:
thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub-interval,
4519:
1112:
relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are
7311:
10786:
4286:
4074:
2477:
10534:
relies on dividing the region under the function into a series of rectangles corresponding to function values and multiplies by the step width to find the sum. A better approach, the
10322:
can also be used, assuming that the integrand can be written as a product of Meijer G-functions. There are also many less common ways of calculating definite integrals; for instance,
9052:
6782:
8317:
1724:
8382:
6515:
8174:
2820:
8426:
7491:
6431:{\displaystyle \left(\int \left|f(x)+g(x)\right|^{p}\,dx\right)^{1/p}\leq \left(\int \left|f(x)\right|^{p}\,dx\right)^{1/p}+\left(\int \left|g(x)\right|^{p}\,dx\right)^{1/p}.}
7382:
13803:
1828:
1366:
10753:
8045:
of the region between the surface defined by the function and the plane that contains its domain. For example, a function in two dimensions depends on two real variables,
7165:{\displaystyle {\begin{aligned}\int _{a}^{c}f(x)\,dx&{}=\int _{a}^{b}f(x)\,dx-\int _{c}^{b}f(x)\,dx\\&{}=\int _{a}^{b}f(x)\,dx+\int _{b}^{c}f(x)\,dx\end{aligned}}}
2846:
13246:
4297:
9698:
10326:
can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see
8119:
8024:
2047:
9489:
9000:
6787:
The first convention is necessary in consideration of taking integrals over subintervals of ; the second says that an integral taken over a degenerate interval, or a
2898:
1329:
1079:, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.
7838:
5504:
10017:
9988:
1778:
are called the limits (or bounds) of integration, and the integral is said to be over the interval , called the interval of integration. A function is said to be
91:
3845:. A function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of being easier to define than Riemann integrals.
14133:
10057:
10037:
9718:
1386:
9869:
2878:
3048:
my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.
10385:
A major mathematical difficulty in symbolic integration is that in many cases, a relatively simple function does not have integrals that can be expressed in
5308:
7713:
3000:
When the chosen tags are the maximum (respectively, minimum) value of the function in each interval, the
Riemann sum becomes an upper (respectively, lower)
10370:
have been compiled and published over the years for this purpose. With the spread of computers, many professionals, educators, and students have turned to
2907:
9257:
8322:
This reduces the problem of computing a double integral to computing one-dimensional integrals. Because of this, another notation for the integral over
6819:
1629:, which are used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.
13453:
8581:. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly
6132:{\displaystyle \left|\int f(x)g(x)\,dx\right|\leq \left(\int \left|f(x)\right|^{p}\,dx\right)^{1/p}\left(\int \left|g(x)\right|^{q}\,dx\right)^{1/q}.}
5196:
4904:
1854:
1527:
continuous functions are
Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of
1098:. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an
13796:
13470:
5061:
2008:
integrals are required to find exact and rigorous values for these quantities. In each case, one may divide the sought quantity into infinitely many
3398:
1605:
used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with
4760:
4524:
that is compatible with linear combinations. In this situation, the linearity holds for the subspace of functions whose integral is an element of
3949:
integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both
10437:
Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the
8434:
12340:
Rich, Albert; Scheibe, Patrick; Abbasi, Nasser (16 December 2018), "Rule-based integration: An extensive system of symbolic integration rules",
11939:
1591:; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by
12569:
12095:
3506:
12152:
10366:
Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive
7616:
13789:
5804:{\displaystyle \left(\int _{a}^{b}(fg)(x)\,dx\right)^{2}\leq \left(\int _{a}^{b}f(x)^{2}\,dx\right)\left(\int _{a}^{b}g(x)^{2}\,dx\right).}
13932:
9101:. Differential forms are organized by degree. For example, a one-form is a weighted sum of the differentials of the coordinates, such as:
292:
50:
10496:
7707:
If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity:
10629:. For each new step size, only half the new function values need to be computed; the others carry over from the previous size. It then
10457:
and so on). Extending Risch's algorithm to include such functions is possible but challenging and has been an active research subject.
3833:
Although the
Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including:
13570:
10318:
and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution using
8893:
1268:, who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians
17:
12643:
9758:
13839:
13270:
12936:
12841:
12576:
6620:
2282:
13947:
8710:
46:
7213:
is first integrated and then differentiated, the original function is retrieved. An important consequence, sometimes called the
5023:. In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if
1851:
when only the simple
Riemann integral is being used, or the exact type of integral is immaterial. For instance, one might write
10663:
The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives as
10487:-finite function as the solution of a differential equation. This theory also allows one to compute the definite integral of a
10065:
9503:
14203:
12476:
12425:
12390:
12331:
12191:
12142:
12075:
11918:
11895:
11855:
11832:
10701:
1395:
The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of
9107:
3223:
1067:, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of
12846:
12821:
2658:
558:
533:
12408:(English translation by L. C. Young. With two additional notes by Stefan Banach. Second revised ed.), New York: Dover
13402:
13053:
12919:
12671:
9406:
8597:. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that
13328:
12751:
12311:
10491:-function as the sum of a series given by the first coefficients and provides an algorithm to compute any coefficient.
8623:
3871:
3856:
1034:
597:
13465:
11931:
10206:
9877:
7506:
3325:
1139:
later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a
115:
13937:
13384:
12914:
12836:
12498:
12451:
12264:
11979:
11876:
8685:, the total work done by the field on the object is obtained by summing up the differential work done in moving from
1980:
to express the linearity of the integral, a property shared by the
Riemann integral and all generalizations thereof.
1421:
553:
271:
4003:
3971:, an extension of the Lebesgue integral to a more general class of functions, namely, those with a domain that is a
13622:
12851:
12816:
12161:
10944:{\displaystyle \int _{0}^{\pi }\sin(x)\,dx=-\cos(x){\big |}_{x=0}^{x=\pi }=-\cos(\pi )-{\big (}-\cos(0){\big )}=2.}
10716:
Kempf, Jackson and
Morales demonstrated mathematical relations that allow an integral to be calculated by means of
10304:
4477:
538:
9619:
An integration that is performed not over a variable (or, in physics, over a space or time dimension), but over a
4217:{\displaystyle \int _{a}^{b}(\alpha f+\beta g)(x)\,dx=\alpha \int _{a}^{b}f(x)\,dx+\beta \int _{a}^{b}g(x)\,dx.\,}
2614:{\displaystyle a=x_{0}\leq t_{1}\leq x_{1}\leq t_{2}\leq x_{2}\leq \cdots \leq x_{n-1}\leq t_{n}\leq x_{n}=b.\,\!}
14198:
14159:
14020:
13702:
13629:
12831:
12299:
10777:
10410:
10160:
7247:
7196:
1838:
1480:
1460:
1109:
874:
548:
523:
205:
6582:
within intervals where an interval with a higher index lies to the right of one with a lower index. The values
4248:
1782:
if its integral over its domain is finite. If limits are specified, the integral is called a definite integral.
13844:
13612:
13364:
13344:
12766:
12726:
10565:
10545:
Riemann sums, the trapezoidal rule, and Simpson's rule are examples of a family of quadrature rules called the
3893:
3849:
1155:
12013:
Gonzalez, Ivan; Jiu, Lin; Moll, Victor H. (1 January 2020), "An extension of the method of brackets. Part 2",
9008:
6730:
64:
of the region bounded by its graph and the horizontal axis; in the above graph as an example, the integral of
13854:
13617:
13563:
13460:
13394:
13298:
13181:
12826:
12607:
12537:
12303:
12200:
Kempf, Achim; Jackson, David M.; Morales, Alejandro H. (2015), "How to (path-)integrate by differentiating",
8230:
5812:
656:
603:
484:
14110:
3852:, an extension of the Riemann integral which integrates with respect to a function as opposed to a variable.
1675:
13869:
11007:
Integral calculus is a very well established mathematical discipline for which there are many sources. See
10468:
9644:
8590:
8332:
7230:
6469:
1740:
310:
282:
13005:
3311:
is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of
1151:; it is more general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable.
393:
14076:
14005:
13942:
13448:
13318:
12990:
12789:
12532:
10292:
9588:
8127:
907:
515:
353:
325:
13889:
13864:
13849:
13412:
13354:
13211:
12959:
12926:
12794:
12413:
12086:
10522:
Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method, Gaussian quadrature
10300:
8593:
vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on
6578:
are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating
5824:
3954:
2799:
778:
742:
519:
398:
287:
277:
8402:
3964:, a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953.
1503:
While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of
30:
This article is about the concept of definite integrals in calculus. For the indefinite integral, see
13879:
13874:
13417:
13349:
12273:
10675:
The area of an arbitrary two-dimensional shape can be determined using a measuring instrument called
10546:
10454:
10143:
1564:
1484:
1128:
1120:
542:
39:
13369:
5941:
4397:{\displaystyle \int _{E}(\alpha f+\beta g)\,d\mu =\alpha \int _{E}f\,d\mu +\beta \int _{E}g\,d\mu .}
4242:
is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral
3738:
3585:
3384:
In that case, the integral is, as in the Riemannian case, the difference between the area above the
1791:
1334:
1131:
in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of
378:
13975:
13556:
13443:
13422:
13359:
12804:
12082:(Originally published by Cambridge University Press, 1897, based on J. L. Heiberg's Greek version.)
10731:
10692:
10197:
4430:
1437:
1143:
region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century,
677:
237:
2825:
14066:
13970:
13965:
13224:
13046:
12664:
12403:
10664:
10446:
10427:
10406:
10371:
8971:
8566:. Various different line integrals are in use. In the case of a closed curve it is also called a
7955:
of proper integrals as one endpoint of the interval of integration approaches either a specified
7444:
1273:
1175:
1140:
991:
783:
672:
11908:
9670:
7338:
3068:
into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of
1479:
The major advance in integration came in the 17th century with the independent discovery of the
13907:
13734:
13634:
13374:
13303:
13196:
13176:
12544:
10645:
10386:
10323:
10311:
10159:
The most basic technique for computing definite integrals of one real variable is based on the
9648:
9614:
9090:
8766:
The definition of surface integral relies on splitting the surface into small surface elements.
8594:
8559:
8068:
7941:{\displaystyle \int _{a}^{b}f(x)\,dx=\lim _{\varepsilon \to 0}\int _{a+\epsilon }^{b}f(x)\,dx.}
7217:, allows one to compute integrals by using an antiderivative of the function to be integrated.
6591:
6460:
4625:
4614:
3022:
1452:
1429:
1400:
1163:
1027:
956:
917:
801:
737:
661:
12633:
12527:
12321:
12254:
7988:
1298:. Alhazen determined the equations to calculate the area enclosed by the curve represented by
13727:
13722:
13686:
13682:
13607:
13580:
13201:
13093:
12964:
12866:
10760:
10569:
10561:
10527:
10513:
9824:
9576:
9474:
9094:
9080:
8985:
8775:
6808:
6595:
4427:
4424:
4232:
3920:
3875:
3073:
2883:
2761:
of such a tagged partition is the width of the largest sub-interval formed by the partition,
2399:
2378:
2028:
1540:
1456:
1407:, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of
1396:
1301:
1171:
1001:
667:
438:
383:
344:
250:
14128:
14028:
13917:
13912:
13754:
13659:
13379:
13265:
13150:
12811:
12701:
12349:
12223:
10721:
10680:
10398:
10361:
10296:
7967:. In more complicated cases, limits are required at both endpoints, or at interior points.
6202:
3017:
1650:
1433:
1198:
1148:
1006:
986:
912:
581:
500:
474:
388:
9993:
9964:
4064:
on this vector space. Thus, the collection of integrable functions is closed under taking
2003:
from 0 to 1, with 5 yellow right endpoint partitions and 10 green left endpoint partitions
1833:
the integral is called an indefinite integral, which represents a class of functions (the
67:
8:
13834:
13821:
13749:
13739:
13692:
13535:
13323:
13206:
13145:
13114:
12946:
12861:
12856:
12746:
12618:
12235:
12061:
12003:
10756:
10641:
10575:
10553:
Newton–Cotes quadrature rule approximates the polynomial on each subinterval by a degree
10390:
10334:
9624:
9602:
9552:
8961:
8682:
8586:
8221:
7952:
7693:
7210:
6803:
is integrable on any subinterval , but in particular integrals have the property that if
5820:
4408:
3301:
1516:
1187:
1091:
981:
951:
941:
828:
682:
479:
335:
218:
213:
13548:
12353:
12227:
12065:
2015:
As another example, to find the area of the region bounded by the graph of the function
1104:, a function whose derivative is the given function; in this case, they are also called
14193:
14154:
14120:
14000:
13829:
13771:
13654:
13505:
13407:
13313:
13260:
13186:
13119:
13039:
12969:
12906:
12799:
12771:
12736:
12657:
12440:
12367:
12239:
12213:
12131:
12050:
12022:
11956:
10988:
10968:
10725:
10367:
10349:
10348:
Specific results which have been worked out by various techniques are collected in the
10042:
10022:
9836:
9703:
9636:
9572:
9568:
9564:
9560:
4068:, and the integral of a linear combination is the linear combination of the integrals:
4065:
3994:
3039:
introduced the integral bearing his name, explaining this integral thus in a letter to
1371:
946:
849:
833:
773:
768:
763:
727:
608:
527:
433:
428:
232:
227:
13781:
9842:
7810:{\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\int _{a}^{b}f(x)\,dx.}
3841:, which is defined by Darboux sums (restricted Riemann sums) yet is equivalent to the
2851:
14149:
14038:
13985:
13744:
13707:
13293:
13018:
12995:
12931:
12901:
12893:
12871:
12741:
12494:
12472:
12447:
12421:
12386:
12327:
12307:
12260:
12243:
12187:
12138:
12114:
12071:
12054:
12042:
11975:
11927:
11914:
11891:
11872:
11851:
11828:
11105:
10976:
10962:
10539:
10461:
10442:
10438:
10394:
10342:
10327:
10319:
9598:
9086:
9070:
8887:
at each point, which will give a scalar field, which is integrated over the surface:
8878:
8783:
8062:
7976:
7611:
7601:
5298:
4061:
3939:
3938:, which is a kind of Riemann–Stieltjes integral with respect to certain functions of
3905:
1206:
1020:
854:
632:
510:
463:
320:
315:
12416:(2008), "Henri Lebesgue", in Timothy Gowers; June Barrow-Green; Imre Leader (eds.),
12399:
12371:
12109:
8029:
Just as the definite integral of a positive function of one variable represents the
2990:{\displaystyle \left|S-\sum _{i=1}^{n}f(t_{i})\,\Delta _{i}\right|<\varepsilon .}
14172:
13957:
13922:
13859:
13812:
13697:
13677:
13495:
13434:
13255:
13155:
13110:
13098:
13013:
12876:
12761:
12721:
12716:
12711:
12706:
12696:
12357:
12250:
12231:
12104:
12032:
11864:
11847:
Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra
11097:
10535:
10531:
10501:
10423:
10379:
10338:
9664:
9058:
8967:
8949:
8945:
8529:
7701:
6788:
4663:
4598:
4590:
4437:
3968:
3961:
3867:
3842:
3838:
3077:
3005:
3001:
2454:
1552:
1528:
1520:
1404:
1222:
1167:
1136:
1113:
864:
758:
732:
593:
505:
469:
12090:
10430:, does just that for functions and antiderivatives built from rational functions,
9720:
is the radius. In the case of a simple disc created by rotating a curve about the
9597:. Summations and integrals can be put on the same foundations using the theory of
9393:{\displaystyle G(x,y,z)\,dx\wedge dy+E(x,y,z)\,dy\wedge dz+F(x,y,z)\,dz\wedge dx.}
6937:{\displaystyle \int _{a}^{b}f(x)\,dx=\int _{a}^{c}f(x)\,dx+\int _{c}^{b}f(x)\,dx.}
1197:
The first documented systematic technique capable of determining integrals is the
1154:
Integrals may be generalized depending on the type of the function as well as the
14105:
14048:
13927:
13712:
13649:
13644:
13639:
13000:
12883:
12756:
12486:
12464:
12287:
11992:
10982:
10578:
halves the step widths incrementally, giving trapezoid approximations denoted by
10414:
9659:
9640:
8598:
8186:
7820:
If the integrand is only defined or finite on a half-open interval, for instance
7416:
2468:
2272:
it will reach a limit which is the exact value of the area sought (in this case,
1973:{\textstyle \int _{a}^{b}(c_{1}f+c_{2}g)=c_{1}\int _{a}^{b}f+c_{2}\int _{a}^{b}g}
1638:
1568:
1512:
1508:
1504:
996:
869:
823:
818:
705:
618:
563:
12954:
10483:-finite function. This provides an algorithm to express the antiderivative of a
8762:
6438:
An analogue of this inequality for Lebesgue integral is used in construction of
3916:
1147:
generalized Riemann's formulation by introducing what is now referred to as the
13308:
13124:
12435:
12209:
12183:
12175:
11988:
11967:
11951:
11904:
11841:
10471:
with polynomial coefficients. Most of the elementary and special functions are
10450:
10139:
9620:
8678:
7420:
5605:{\displaystyle \left|\int _{a}^{b}f(x)\,dx\right|\leq \int _{a}^{b}|f(x)|\,dx.}
5302:
3950:
3935:
3928:
3924:
3909:
3036:
1834:
1592:
1532:
1202:
1144:
1100:
879:
687:
454:
31:
12562:
12277:
11101:
10679:. The volume of irregular objects can be measured with precision by the fluid
9248:
measure infinitesimal oriented lengths parallel to the three coordinate axes.
8387:
Integration over more general domains is possible. The integral of a function
8224:
states that this integral can be expressed as an equivalent iterated integral
3886:
14187:
13990:
13716:
13672:
13520:
13515:
13500:
13490:
13191:
13105:
13080:
12985:
12594:
12548:
12279:
Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern. Erster Band
12118:
12046:
11845:
11109:
10630:
10315:
10146:
is used to calculate the difference in free energy between two given states.
9496:
9492:
8801:
For an example of applications of surface integrals, consider a vector field
8787:
8525:
5816:
4582:
4556:
4457:. Then one may define an abstract integration map assigning to each function
3989:
The collection of Riemann-integrable functions on a closed interval forms a
3897:
3882:
3485:{\displaystyle \int _{E}f\,d\mu =\int _{E}f^{+}\,d\mu -\int _{E}f^{-}\,d\mu }
2009:
1548:
1544:
1159:
1132:
859:
623:
373:
330:
4639:
and can be generalized to other notions of integral (Lebesgue and Daniell).
4601:
to functions with values in a locally compact topological vector space. See
3885:, used for integration on locally compact topological groups, introduced by
2067:), then construct rectangles using the right end height of each piece (thus
1988:
13277:
13076:
12731:
12623:, University of Oldenburg. A new concept to an old problem. Online textbook
12581:, CIT, an online textbook that includes a complete introduction to calculus
12378:
12291:
12037:
12008:, Freeman, Alexander (trans.), Cambridge University Press, pp. 200–201
11943:
10518:
8795:
8791:
8669:
8578:
8574:
8544:
8509:{\displaystyle \int _{D}f(\mathbf {x} )d^{n}\mathbf {x} \ =\int _{D}f\,dV.}
3990:
3972:
3901:
3860:
3297:
1602:
1495:, whose notation for integrals is drawn directly from the work of Leibniz.
1488:
1448:
1295:
1226:
1166:
of integration is replaced by a curve connecting two points in space. In a
1124:
613:
358:
11085:
9990:
is the velocity expressed as a function of time. The work done by a force
3290:
where the integral on the right is an ordinary improper Riemann integral (
2431:
1511:
memorably attacked the vanishing increments used by Newton, calling them "
45:"Area under the curve" redirects here. For the pharmacology integral, see
12460:
12126:
10698:
10419:
10178:
to be integrated over a given interval . Then, find an antiderivative of
9471:
measure oriented areas parallel to the coordinate two-planes. The symbol
9076:
8868:
8190:
7956:
6456:
4667:
4622:
4228:
3040:
2460:
1464:
1269:
1250:
1087:
1048:
976:
61:
12362:
10985: – Mathematical symbol used to denote integrals and antiderivatives
14084:
13995:
13980:
13602:
13530:
13221:
12611:
12600:
10717:
10676:
9820:
9643:
falling within a certain range. Moreover, the integral under an entire
8582:
7981:
7206:
6792:
5468:{\displaystyle (fg)(x)=f(x)g(x),\;f^{2}(x)=(f(x))^{2},\;|f|(x)=|f(x)|.}
4618:
3946:
3004:, suggesting the close connection between the Riemann integral and the
1599:
of the French Academy around 1819–1820, reprinted in his book of 1822.
1455:, who provided the first hints of a connection between integration and
1389:
1368:
in contemporary notation), for any given non-negative integer value of
1246:
1218:
1210:
1072:
722:
646:
368:
363:
267:
12585:
9567:
simultaneously generalizes the three theorems of vector calculus: the
7680:{\displaystyle \int _{0}^{\infty }{\frac {dx}{(x+1){\sqrt {x}}}}=\pi }
3863:, which generalizes both the Riemann–Stieltjes and Lebesgue integrals.
1075:. Integration was initially used to solve problems in mathematics and
13902:
13525:
13510:
11496:
10568:, in which the integrand is approximated by expanding it in terms of
10402:
9594:
1524:
1471:
to a general power, including negative powers and fractional powers.
1441:
1095:
1056:
651:
641:
12507:
11084:
Dennis, David; Kreinovich, Vladik; Rump, Siegfried M. (1998-05-01).
10200:
on the path of integration, by the fundamental theorem of calculus,
7606:
13759:
13592:
13165:
13134:
13085:
13062:
12627:
12218:
12027:
10431:
9832:
9658:
of a two-dimensional region that has a curved boundary, as well as
9556:
7697:
6795:. One reason for the first convention is that the integrability of
6439:
3300:
improper Riemann integral). For a suitable class of functions (the
1492:
1242:
1068:
717:
105:
56:
10310:
Alternative methods exist to compute more complex integrals. Many
2012:
pieces, then sum the pieces to achieve an accurate approximation.
14010:
13897:
10979: – Equations with an unknown function under an integral sign
10375:
3997:
and multiplication by a scalar, and the operation of integration
1280:
1265:
1238:
1123:, the principles of integration were formulated independently by
1076:
35:
12129:; Nash, Stephen (1989), "Chapter 5: Numerical Quadrature",
10652:-point Gaussian method is exact for polynomials of degree up to
10526:
Definite integrals may be approximated using several methods of
8534:
5277:{\displaystyle \int _{c}^{d}f(x)\,dx\leq \int _{a}^{b}f(x)\,dx.}
4985:{\displaystyle \int _{a}^{b}f(x)\,dx\leq \int _{a}^{b}g(x)\,dx.}
3296:
is a strictly decreasing positive function, and therefore has a
1519:. Integration was first rigorously formalized, using limits, by
10705:
9098:
8934:{\displaystyle \int _{S}{\mathbf {v} }\cdot \,d{\mathbf {S} }.}
8867:
in unit amount of time. To find the flux, one need to take the
8185:
indicates that integration is taken with respect to area. This
8042:
5142:{\displaystyle \int _{a}^{b}f(x)\,dx<\int _{a}^{b}g(x)\,dx.}
4897:
is bounded above by the upper and lower sums, respectively, of
1580:
1254:
1234:
1230:
1064:
10542:
approximates the integrand by a piecewise quadratic function.
9635:
Integrals are used extensively in many areas. For example, in
3072:". The definition of the Lebesgue integral thus begins with a
1515:". Calculus acquired a firmer footing with the development of
12160:(M.A. thesis), University of British Columbia, archived from
11715:
10196:
on the interval. Provided the integrand and integral have no
8779:
8602:
8563:
6704:{\displaystyle \int _{a}^{b}f(x)\,dx=-\int _{b}^{a}f(x)\,dx.}
4613:
A number of general inequalities hold for Riemann-integrable
4528:(i.e. "finite"). The most important special cases arise when
1261:
1119:
Although methods of calculating areas and volumes dated from
51:
Receiver operating characteristic § Area under the curve
13031:
12649:
10771:
10633:
a polynomial through the approximations, and extrapolate to
10413:, and the operations of multiplication and composition. The
9651:
with no negative values could be a density function or not.
6150:, Hölder's inequality becomes the Cauchy–Schwarz inequality.
4846:{\displaystyle m(b-a)\leq \int _{a}^{b}f(x)\,dx\leq M(b-a).}
2334:{\displaystyle \int _{0}^{1}{\sqrt {x}}\,dx={\frac {2}{3}},}
60:
A definite integral of a function can be represented as the
14089:
14061:
14056:
14033:
12320:
Montesinos, Vicente; Zizler, Peter; Zizler, Václav (2015),
11727:
10019:(given as a function of position) from an initial position
9828:
9655:
8858:
8752:{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {s} .}
8030:
6530:. This means that the upper and lower sums of the function
3807:
3651:
1563:
The notation for the indefinite integral was introduced by
1162:
is defined for functions of two or more variables, and the
1060:
11958:
An introduction to probability theory and its applications
9639:, integrals are used to determine the probability of some
2359:, multiplied by the infinitesimal step widths, denoted by
13578:
12644:
Evaluation of Definite Integrals by Symbolic Manipulation
10128:{\displaystyle W_{A\rightarrow B}=\int _{A}^{B}F(x)\,dx.}
9540:{\displaystyle E\mathbf {i} +F\mathbf {j} +G\mathbf {k} }
9002:
in the complex plane, the integral is denoted as follows
5920:
are two Riemann-integrable functions. Then the functions
12646:(1972) — a cookbook of definite integral techniques
10780:
allows straightforward calculations of basic functions:
8193:, and represents the (signed) volume under the graph of
8033:
of the region between the graph of the function and the
3052:
As Folland puts it, "To compute the Riemann integral of
1158:
over which the integration is performed. For example, a
93:
is the yellow (−) area subtracted from the blue (+) area
13811:
11972:
Real Analysis: Modern Techniques and Their Applications
9839:, the displacement of an object over the time interval
8617:, may be expressed (in terms of vector quantities) as:
8519:
8041:
of a positive function of two variables represents the
7320:
is continuous on , differentiable on the open interval
5819:
theory, where the left hand side is interpreted as the
4992:
This is a generalization of the above inequalities, as
1467:
generalized Cavalieri's method, computing integrals of
1279:
In the Middle East, Hasan Ibn al-Haytham, Latinized as
12319:
11823:
Anton, Howard; Bivens, Irl C.; Davis, Stephen (2016),
11502:
9213:{\displaystyle E(x,y,z)\,dx+F(x,y,z)\,dy+G(x,y,z)\,dz}
8982:. When a complex function is integrated along a curve
4291:
is a linear functional on this vector space, so that:
3280:{\displaystyle \int f=\int _{0}^{\infty }f^{*}(t)\,dt}
3114:" philosophy, the integral of a non-negative function
2652:
with respect to such a tagged partition is defined as
2107:
2031:
1857:
1845:
In advanced settings, it is not uncommon to leave out
13227:
12563:
Elementary Calculus: An Approach Using Infinitesimals
12493:, Princeton, New Jersey: Princeton University Press,
10789:
10734:
10209:
10068:
10045:
10025:
9996:
9967:
9880:
9845:
9761:
9706:
9673:
9506:
9477:
9409:
9260:
9110:
9011:
8988:
8896:
8713:
8626:
8538:
A line integral sums together elements along a curve.
8437:
8405:
8335:
8233:
8130:
8071:
7991:
7841:
7716:
7619:
7509:
7447:
7341:
7250:
6956:
6822:
6733:
6623:
6472:
6211:
5950:
5632:
5507:
5311:
5199:
5064:
4907:
4763:
4480:
4300:
4251:
4077:
4006:
3504:
3401:
3328:
3226:
3132:
of the areas between a thin horizontal strip between
2910:
2886:
2854:
2828:
2802:
2719:{\displaystyle \sum _{i=1}^{n}f(t_{i})\,\Delta _{i};}
2661:
2480:
2285:
2106:
1794:
1678:
1447:
Further steps were made in the early 17th century by
1374:
1337:
1304:
118:
70:
11083:
10958:
9589:
Summation § Approximation by definite integrals
8861:
is defined as the quantity of fluid flowing through
5007:
is the integral of the constant function with value
2348:
is the result of a weighted sum of function values,
12199:
11936:
Numerical Methods in Scientific Computing, Volume I
11805:
9464:{\displaystyle dx\wedge dy,dz\wedge dx,dy\wedge dz}
8774:generalizes double integrals to integration over a
7175:is then well-defined for any cyclic permutation of
4605:for an axiomatic characterization of the integral.
1531:—to which Riemann's definition does not apply, and
13240:
12439:
12130:
11955:
10943:
10747:
10279:
10127:
10051:
10031:
10011:
9982:
9950:
9863:
9819:Integrals are also used in physics, in areas like
9811:
9712:
9692:
9539:
9483:
9463:
9392:
9212:
9064:
9046:
8994:
8933:
8751:
8651:
8508:
8420:
8376:
8311:
8168:
8113:
8018:
7940:
7832:, then again a limit may provide a finite result:
7809:
7687:has unbounded intervals for both domain and range.
7679:
7579:
7485:
7376:
7305:
7229:be a continuous real-valued function defined on a
7190:
7164:
6947:With the first convention, the resulting relation
6936:
6776:
6703:
6509:
6430:
6131:
5803:
5604:
5467:
5276:
5141:
4984:
4845:
4513:
4396:
4280:
4216:
4049:
3817:
3484:
3373:
3279:
2989:
2892:
2872:
2840:
2814:
2718:
2632:, each of which is "tagged" with a specific point
2613:
2333:
2260:
2041:
1972:
1822:
1718:
1380:
1360:
1323:
1090:of the region in the plane that is bounded by the
190:
85:
12412:
12339:
11721:
11349:
10711:
9667:using the equation for the volume of a cylinder,
9647:must equal 1, which provides a test of whether a
8652:{\displaystyle W=\mathbf {F} \cdot \mathbf {s} .}
4407:More generally, consider the vector space of all
4231:-valued Lebesgue-integrable functions on a given
2610:
14185:
12381:(1987), "Chapter 1: Abstract Integration",
12124:
11822:
11793:
11781:
11769:
11697:
11637:
11625:
11613:
11601:
11589:
11577:
11565:
11553:
11538:
11313:
11289:
11012:
10644:evaluates the function at the roots of a set of
10460:More recently a new approach has emerged, using
10280:{\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}
9951:{\displaystyle x(b)-x(a)=\int _{a}^{b}v(t)\,dt,}
8836:is a vector. Imagine that a fluid flows through
7985:Double integral computes volume under a surface
7880:
7755:
7580:{\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}
3691:
3541:
3374:{\displaystyle \int _{E}|f|\,d\mu <+\infty .}
2063:, one can divide the interval into five pieces (
1749:, indicates that the variable of integration is
1260:A similar method was independently developed in
1221:in the 3rd century BC and used to calculate the
191:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
12012:
11997:, Chez Firmin Didot, père et fils, p. §231
11926:
11757:
11745:
11733:
4593:for the case of real-valued functions on a set
2097:) and sum their areas to get the approximation
1641:in 1690: "Ergo et horum Integralia aequantur".
12628:Difference Equations to Differential Equations
12570:A Brief Introduction to Infinitesimal Calculus
9812:{\displaystyle \pi \int _{a}^{b}f^{2}(x)\,dx.}
8978:instead of a real function of a real variable
6201:are also Riemann-integrable and the following
5297:are two functions, then we may consider their
4050:{\displaystyle f\mapsto \int _{a}^{b}f(x)\;dx}
3904:, and (most elegantly, as the gauge integral)
1094:of a given function between two points in the
13797:
13564:
13047:
12665:
12459:
12096:Bulletin of the American Mathematical Society
11910:A History Of Mathematical Notations Volume II
10930:
10905:
10851:
9251:A differential two-form is a sum of the form
7209:and integration are inverse operations: if a
4514:{\displaystyle f\mapsto \int _{E}f\,d\mu ,\,}
1028:
3718:
3694:
3565:
3544:
2459:The Riemann integral is defined in terms of
1249:of revolution, the volume of a segment of a
12085:
11888:The History of Mathematics: An Introduction
9089:is a mathematical concept in the fields of
7306:{\displaystyle F(x)=\int _{a}^{x}f(t)\,dt.}
4602:
3923:, which define integration with respect to
3870:, which subsumes the Lebesgue integral and
1644:
1294: AD) derived a formula for the sum of
1192:
13804:
13790:
13571:
13557:
13471:Vitale's random Brunn–Minkowski inequality
13054:
13040:
12672:
12658:
9593:The discrete equivalent of integration is
5414:
5363:
5287:Products and absolute values of functions.
4563:is a finite-dimensional vector space over
4281:{\displaystyle f\mapsto \int _{E}f\,d\mu }
4040:
2445:Riemann integrals and Lebesgue integrals.
1035:
1021:
12434:
12361:
12217:
12180:A History of Mathematics: An Introduction
12108:
12036:
12026:
11182:
10820:
10772:Using the fundamental theorem of calculus
10301:integration by trigonometric substitution
10237:
10115:
9938:
9799:
9371:
9328:
9285:
9203:
9169:
9135:
9034:
8917:
8790:. The function to be integrated may be a
8496:
8408:
8364:
8299:
8287:
8159:
7928:
7869:
7797:
7744:
7537:
7293:
7151:
7114:
7068:
7031:
6988:
6924:
6887:
6850:
6761:
6691:
6651:
6500:
6399:
6332:
6265:
6100:
6036:
5983:
5786:
5735:
5675:
5592:
5540:
5264:
5227:
5129:
5092:
4972:
4935:
4893:then each of the upper and lower sums of
4812:
4510:
4500:
4384:
4358:
4332:
4271:
4213:
4203:
4163:
4123:
3475:
3445:
3415:
3352:
3270:
2962:
2702:
2609:
2308:
1837:) whose derivative is the integrand. The
1810:
1706:
1551:of an infinite Riemann sum, based on the
1459:. Barrow provided the first proof of the
1351:
151:
12937:Common integrals in quantum field theory
12608:Elementary Treatise on Integral Calculus
12286:
11863:
11827:(11th ed.), John Wiley & Sons,
11385:
11373:
10517:
10434:, logarithm, and exponential functions.
9654:Integrals can be used for computing the
9047:{\displaystyle \int _{\gamma }f(z)\,dz.}
8853:determines the velocity of the fluid at
8761:
8533:
7980:
7605:
6777:{\displaystyle \int _{a}^{a}f(x)\,dx=0.}
4727:are therefore bounded by, respectively,
3021:
2467:of an interval. A tagged partition of a
1987:
1632:
1170:, the curve is replaced by a piece of a
55:
12847:Differentiation under the integral sign
12590:, Fullerton College, an online textbook
12491:A Source Book in Mathematics, 1200-1800
12385:(International ed.), McGraw-Hill,
12276:(1899), Gerhardt, Karl Immanuel (ed.),
12272:
12001:
11987:
11974:(2nd ed.), John Wiley & Sons,
11966:
11932:"Chapter 5: Numerical Integration"
11840:
11709:
11685:
11673:
11526:
11514:
11490:
11478:
11451:
11427:
11397:
11361:
11301:
11277:
11238:
11222:
11086:"Intervals and the Origins of Calculus"
11008:
9608:
8573:The function to be integrated may be a
8312:{\displaystyle \int _{a}^{b}\left\,dx.}
7970:
7415:be a real-valued function defined on a
6171:are Riemann-integrable functions. Then
1637:The term was first printed in Latin by
1428:= −1 required the invention of a
559:Differentiating under the integral sign
47:Area under the curve (pharmacokinetics)
14:
14186:
12485:
12249:
12150:
11950:
11903:
11885:
11661:
11649:
11325:
11262:
11250:
11234:
11218:
11170:
11146:
11035:
10702:compass-and-straightedge constructions
8562:to be integrated is evaluated along a
8216:. Under suitable conditions (e.g., if
7951:That is, the improper integral is the
7215:second fundamental theorem of calculus
5940:are also integrable and the following
3304:) this defines the Lebesgue integral.
2471:on the real line is a finite sequence
1719:{\displaystyle \int _{a}^{b}f(x)\,dx.}
1558:
1536:
13785:
13552:
13035:
12653:
12377:
12202:Journal of Physics: Conference Series
12060:
11549:
11547:
11466:
11462:
11460:
11439:
11423:
11421:
11412:
11408:
11406:
11337:
11273:
11271:
11047:
9755:, the volume of the disc is equal to:
9737:, and its height is the differential
8377:{\displaystyle \iint _{R}f(x,y)\,dA.}
7595:
6510:{\displaystyle \int _{a}^{b}f(x)\,dx}
4544:, or a finite extension of the field
3110:Using the "partitioning the range of
2421:
1474:
13484:Applications & related
12638:Holistic Numerical Methods Institute
12398:
12174:
11882:. In particular chapters III and IV.
11503:Montesinos, Zizler & Zizler 2015
11206:
11194:
11158:
11134:
11122:
11071:
11059:
8955:
8520:Line integrals and surface integrals
4711:. Since the lower and upper sums of
3011:
2848:such that, for any tagged partition
1766:is called the integrand, the points
13403:Marcinkiewicz interpolation theorem
8169:{\displaystyle \int _{R}f(x,y)\,dA}
7704:on progressively larger intervals.
6602:. Integrals can also be defined if
2624:This partitions the interval into
2448:
2405:Darboux lower sums of the function
2384:Darboux upper sums of the function
1785:When the limits are omitted, as in
1733:represents integration. The symbol
24:
13329:Symmetric decreasing rearrangement
13233:
12469:Introduction to Numerical Analysis
12418:Princeton Companion to Mathematics
12326:(illustrated ed.), Springer,
12323:An Introduction to Modern Analysis
12154:A History of the Definite Integral
11544:
11457:
11418:
11403:
11268:
10759:, allowing them to be computed by
10736:
10564:. One solution to this problem is
8662:For an object moving along a path
8391:, with respect to volume, over an
7765:
7727:
7630:
3828:
3365:
3246:
2964:
2704:
1983:
1331:(which translates to the integral
100:Part of a series of articles about
49:. For the statistics concept, see
27:Operation in mathematical calculus
25:
14215:
12520:
12467:(2002), "Topics in Integration",
11806:Kempf, Jackson & Morales 2015
8053:, and the integral of a function
7406:
7237:be the function defined, for all
5620:are both Riemann-integrable then
4666:on that interval. Thus there are
2815:{\displaystyle \varepsilon >0}
1253:of revolution, and the area of a
12634:Numerical Methods of Integration
12606:Johnson, William Woolsey (1909)
12091:"Integration in abstract spaces"
11994:Théorie analytique de la chaleur
10961:
10697:Area can sometimes be found via
10305:integration by partial fractions
9743:. Using an integral with bounds
9533:
9522:
9511:
8923:
8909:
8742:
8731:
8642:
8634:
8473:
8455:
8421:{\displaystyle \mathbb {R} ^{n}}
7434:are functions such that for all
7220:
6520:over an interval is defined if
5626:is also Riemann-integrable, and
2430:
2398:
2377:
1663:with respect to a real variable
1537:different definition of integral
1498:
1276:to find the volume of a sphere.
14160:Pearson correlation coefficient
13703:Least-squares spectral analysis
13630:Fundamental theorem of calculus
12555:
12342:Journal of Open Source Software
12302:, vol. 14 (2nd ed.),
12300:Graduate Studies in Mathematics
12110:10.1090/S0002-9904-1953-09694-X
11825:Calculus: Early Transcendentals
11815:
11799:
11787:
11775:
11763:
11751:
11739:
11722:Rich, Scheibe & Abbasi 2018
11703:
11691:
11679:
11667:
11655:
11643:
11631:
11619:
11607:
11595:
11583:
11571:
11559:
11532:
11520:
11508:
11484:
11472:
11445:
11433:
11391:
11379:
11367:
11355:
11343:
11331:
11319:
11307:
11295:
11283:
11256:
11244:
11228:
11212:
11200:
11188:
11176:
11164:
11152:
10778:fundamental theorem of calculus
10475:-finite, and the integral of a
10411:inverse trigonometric functions
10161:fundamental theorem of calculus
9630:
9065:Integrals of differential forms
8782:); it can be thought of as the
8704:. This gives the line integral
8428:is denoted by symbols such as:
7203:fundamental theorem of calculus
7197:Fundamental theorem of calculus
7191:Fundamental theorem of calculus
4856:Inequalities between functions.
4608:
3076:, μ. In the simplest case, the
2788:over the interval is equal to
1839:fundamental theorem of calculus
1481:fundamental theorem of calculus
1461:fundamental theorem of calculus
1245:, the volume of a segment of a
1110:fundamental theorem of calculus
12508:"Arabic mathematical notation"
12420:, Princeton University Press,
12236:10.1088/1742-6596/626/1/012015
12133:Numerical Methods and Software
11794:Kahaner, Moler & Nash 1989
11782:Kahaner, Moler & Nash 1989
11770:Kahaner, Moler & Nash 1989
11698:Anton, Bivens & Davis 2016
11638:Anton, Bivens & Davis 2016
11626:Anton, Bivens & Davis 2016
11614:Anton, Bivens & Davis 2016
11602:Anton, Bivens & Davis 2016
11590:Anton, Bivens & Davis 2016
11578:Anton, Bivens & Davis 2016
11566:Anton, Bivens & Davis 2016
11554:Anton, Bivens & Davis 2016
11539:Anton, Bivens & Davis 2016
11314:Anton, Bivens & Davis 2016
11290:Anton, Bivens & Davis 2016
11140:
11128:
11116:
11077:
11065:
11053:
11041:
11029:
11013:Anton, Bivens & Davis 2016
11001:
10925:
10919:
10897:
10891:
10845:
10839:
10817:
10811:
10755:. This can also be applied to
10720:. Their calculus involves the
10712:Integration by differentiation
10686:
10271:
10265:
10256:
10250:
10234:
10228:
10149:
10112:
10106:
10077:
10006:
10000:
9977:
9971:
9935:
9929:
9905:
9899:
9890:
9884:
9858:
9846:
9796:
9790:
9726:-axis, the radius is given by
9368:
9350:
9325:
9307:
9282:
9264:
9200:
9182:
9166:
9148:
9132:
9114:
9031:
9025:
8778:(which may be a curved set in
8611:, multiplied by displacement,
8459:
8451:
8361:
8349:
8284:
8272:
8156:
8144:
8108:
8096:
8090:
8078:
8013:
8001:
7925:
7919:
7887:
7866:
7860:
7794:
7788:
7762:
7741:
7735:
7658:
7646:
7571:
7565:
7556:
7550:
7534:
7528:
7477:
7471:
7457:
7451:
7371:
7365:
7356:
7350:
7290:
7284:
7260:
7254:
7148:
7142:
7111:
7105:
7065:
7059:
7028:
7022:
6985:
6979:
6921:
6915:
6884:
6878:
6847:
6841:
6758:
6752:
6688:
6682:
6648:
6642:
6497:
6491:
6446:
6385:
6379:
6318:
6312:
6251:
6245:
6236:
6230:
6086:
6080:
6022:
6016:
5980:
5974:
5968:
5962:
5811:This inequality, known as the
5777:
5770:
5726:
5719:
5672:
5666:
5663:
5654:
5588:
5584:
5578:
5571:
5537:
5531:
5458:
5454:
5448:
5441:
5434:
5428:
5424:
5416:
5402:
5398:
5392:
5386:
5380:
5374:
5357:
5351:
5345:
5339:
5330:
5324:
5321:
5312:
5261:
5255:
5224:
5218:
5126:
5120:
5089:
5083:
4969:
4963:
4932:
4926:
4837:
4825:
4809:
4803:
4779:
4767:
4484:
4329:
4311:
4255:
4200:
4194:
4160:
4154:
4120:
4114:
4111:
4093:
4037:
4031:
4010:
3775:
3769:
3753:
3747:
3709:
3703:
3676:
3670:
3619:
3613:
3597:
3591:
3556:
3550:
3526:
3520:
3348:
3340:
3307:A general measurable function
3267:
3261:
2959:
2946:
2867:
2855:
2699:
2686:
1992:Approximations to integral of
1905:
1873:
1823:{\displaystyle \int f(x)\,dx,}
1807:
1801:
1703:
1697:
1669:on an interval is written as
1649:In general, the integral of a
1422:Cavalieri's quadrature formula
1361:{\displaystyle \int x^{k}\,dx}
1055:is the continuous analog of a
185:
179:
170:
164:
148:
142:
80:
74:
34:. For the set of numbers, see
13:
1:
14099:Deep Learning Related Metrics
13299:Convergence almost everywhere
13061:
12679:
12630:, an introduction to calculus
12304:American Mathematical Society
12256:Real Analysis and Foundations
12151:Kallio, Bruce Victor (1966),
12005:The analytical theory of heat
11989:Fourier, Jean Baptiste Joseph
11890:(7th ed.), McGraw-Hill,
11734:Gonzalez, Jiu & Moll 2020
11022:
10748:{\displaystyle \partial _{x}}
10670:
10469:linear differential equations
10467:, which are the solutions of
10154:
9582:
8326:uses a double integral sign:
7590:
6799:on an interval implies that
6534:are evaluated on a partition
3979:
3388:-axis and the area below the
2463:of functions with respect to
1513:ghosts of departed quantities
1284:
1264:around the 3rd century AD by
1059:, which is used to calculate
485:Integral of inverse functions
14204:Linear operators in calculus
12595:Notes on First-Year Calculus
12282:, Berlin: Mayer & Müller
12000:Available in translation as
10683:as the object is submerged.
10507:
9645:probability density function
8585:or, for a vector field, the
5815:, plays a prominent role in
3984:
3056:, one partitions the domain
2841:{\displaystyle \delta >0}
7:
13943:Sensitivity and specificity
13466:Prékopa–Leindler inequality
13319:Locally integrable function
13241:{\displaystyle L^{\infty }}
12752:Lebesgue–Stieltjes integral
12533:Encyclopedia of Mathematics
12442:Mathematics and Its History
12414:Siegmund-Schultze, Reinhard
11758:Dahlquist & Björck 2008
11746:Dahlquist & Björck 2008
10954:
10766:
10479:-finite function is also a
10355:
10333:Computations of volumes of
10293:integration by substitution
10138:Integrals are also used in
8589:of the vector field with a
8558:) is an integral where the
7486:{\displaystyle f(x)=F'(x).}
3872:Lebesgue–Stieltjes integral
3857:Lebesgue–Stieltjes integral
3213:. The Lebesgue integral of
908:Calculus on Euclidean space
326:Logarithmic differentiation
10:
14220:
13212:Square-integrable function
12767:Riemann–Stieltjes integral
12727:Henstock–Kurzweil integral
12545:Online Integral Calculator
12471:(3rd ed.), Springer,
12274:Leibniz, Gottfried Wilhelm
10761:functional differentiation
10690:
10566:Clenshaw–Curtis quadrature
10511:
10359:
9693:{\displaystyle \pi r^{2}h}
9612:
9586:
9495:, which is similar to the
9074:
9068:
8959:
8813:; that is, for each point
8523:
7974:
7599:
7377:{\displaystyle F'(x)=f(x)}
7194:
5491:then the same is true for
3955:fractional Brownian motion
3953:and processes such as the
3894:Henstock–Kurzweil integral
3850:Riemann–Stieltjes integral
3015:
2628:sub-intervals indexed by
2452:
1185:
1181:
44:
29:
14168:
14142:
14119:
14098:
14075:
14047:
14019:
13956:
13888:
13820:
13768:
13668:
13587:
13483:
13461:Minkowski–Steiner formula
13431:
13393:
13337:
13286:
13220:
13164:
13133:
13069:
13006:Proof that 22/7 exceeds π
12978:
12945:
12892:
12780:
12687:
12565:, University of Wisconsin
12383:Real and Complex Analysis
11913:, Open Court Publishing,
11886:Burton, David M. (2011),
10455:incomplete gamma function
10337:can usually be done with
10144:thermodynamic integration
9403:Here the basic two-forms
8114:{\displaystyle R=\times }
5813:Cauchy–Schwarz inequality
5479:is Riemann-integrable on
1565:Gottfried Wilhelm Leibniz
1129:Gottfried Wilhelm Leibniz
1121:ancient Greek mathematics
642:Summand limit (term test)
40:Integral (disambiguation)
18:Integration (mathematics)
13444:Isoperimetric inequality
12578:Sean's Applied Math Book
12002:Fourier, Joseph (1878),
10994:
10693:Quadrature (mathematics)
10428:computer algebra systems
10372:computer algebra systems
9823:to find quantities like
9563:of vector calculus, and
8019:{\displaystyle z=f(x,y)}
6590:, the end-points of the
5189:is non-negative for all
4431:topological vector space
3993:under the operations of
2042:{\textstyle {\sqrt {x}}}
1645:Terminology and notation
1567:in 1675. He adapted the
1193:Pre-calculus integration
321:Implicit differentiation
311:Differentiation notation
238:Inverse function theorem
13971:Calinski-Harabasz index
13449:Brunn–Minkowski theorem
12991:Euler–Maclaurin formula
12067:The Works of Archimedes
11962:, John Wiley & Sons
11850:(2nd ed.), Wiley,
11102:10.1023/A:1009989211143
10665:Monte Carlo integration
10447:hypergeometric function
10407:trigonometric functions
10312:nonelementary integrals
9484:{\displaystyle \wedge }
8995:{\displaystyle \gamma }
8972:complex-valued function
8179:where the differential
7500:is integrable on then
4646:An integrable function
4644:Upper and lower bounds.
4436:over a locally compact
3896:, variously defined by
3859:, further developed by
3128:should be the sum over
2893:{\displaystyle \delta }
2437:Riemann sums converging
1523:. Although all bounded
1324:{\displaystyle y=x^{k}}
1176:three-dimensional space
784:Helmholtz decomposition
14199:Functions and mappings
13635:Calculus of variations
13608:Differential equations
13304:Convergence in measure
13242:
12960:Russo–Vallois integral
12927:Bose–Einstein integral
12842:Parametric derivatives
12601:Understanding Calculus
12405:Theory of the integral
12038:10.1515/math-2020-0062
11930:; Björck, Åke (2008),
11350:Siegmund-Schultze 2008
10945:
10749:
10646:orthogonal polynomials
10523:
10281:
10182:; that is, a function
10129:
10053:
10033:
10013:
9984:
9952:
9865:
9813:
9714:
9694:
9623:, is referred to as a
9615:Functional integration
9555:plays the role of the
9541:
9485:
9465:
9394:
9214:
9091:multivariable calculus
9048:
8996:
8974:of a complex variable
8935:
8767:
8753:
8653:
8539:
8510:
8422:
8378:
8313:
8170:
8115:
8026:
8020:
7942:
7811:
7688:
7681:
7581:
7487:
7378:
7307:
7205:is the statement that
7166:
6938:
6778:
6705:
6511:
6432:
6133:
5861:are two real numbers,
5805:
5606:
5469:
5278:
5143:
4986:
4847:
4515:
4398:
4282:
4227:Similarly, the set of
4218:
4051:
3819:
3486:
3375:
3281:
3050:
3027:
2991:
2942:
2894:
2874:
2842:
2816:
2720:
2682:
2615:
2335:
2262:
2043:
2004:
1974:
1824:
1720:
1401:method of indivisibles
1382:
1362:
1325:
918:Limit of distributions
738:Directional derivative
394:Faà di Bruno's formula
192:
94:
87:
38:. For other uses, see
14134:Intra-list Similarity
13728:Representation theory
13687:quaternionic analysis
13683:Hypercomplex analysis
13581:mathematical analysis
13418:Riesz–Fischer theorem
13243:
13202:Polarization identity
12965:Stratonovich integral
12911:Fermi–Dirac integral
12867:Numerical integration
10946:
10750:
10570:Chebyshev polynomials
10547:Newton–Cotes formulas
10528:numerical integration
10521:
10514:Numerical integration
10314:can be expanded in a
10282:
10130:
10054:
10034:
10014:
9985:
9953:
9866:
9814:
9715:
9695:
9577:Kelvin-Stokes theorem
9542:
9486:
9466:
9395:
9215:
9095:differential topology
9081:Density on a manifold
9049:
8997:
8970:, the integrand is a
8936:
8765:
8754:
8654:
8537:
8511:
8423:
8379:
8314:
8189:can be defined using
8171:
8116:
8021:
7984:
7943:
7812:
7682:
7609:
7582:
7488:
7379:
7308:
7167:
6939:
6779:
6706:
6596:limits of integration
6512:
6433:
6163:is a real number and
6134:
5806:
5607:
5470:
5279:
5144:
4987:
4848:
4516:
4423:, taking values in a
4399:
4283:
4219:
4052:
3921:Stratonovich integral
3874:without depending on
3820:
3487:
3376:
3282:
3045:
3025:
2992:
2922:
2895:
2875:
2843:
2817:
2721:
2662:
2616:
2336:
2263:
2044:
1991:
1975:
1825:
1721:
1633:First use of the term
1383:
1363:
1326:
1002:Mathematical analysis
913:Generalized functions
598:arithmetico-geometric
439:Leibniz integral rule
193:
88:
59:
13660:Table of derivatives
13423:Riesz–Thorin theorem
13266:Infimum and supremum
13225:
13151:Lebesgue integration
12947:Stochastic integrals
12603:, an online textbook
12572:, University of Iowa
12561:Keisler, H. Jerome,
11386:Lieb & Loss 2001
11237:, pp. 249–250;
10787:
10757:functional integrals
10732:
10722:Dirac delta function
10391:elementary functions
10362:Symbolic integration
10335:solids of revolution
10297:integration by parts
10207:
10066:
10043:
10039:to a final position
10023:
10012:{\displaystyle F(x)}
9994:
9983:{\displaystyle v(t)}
9965:
9878:
9843:
9759:
9704:
9671:
9660:computing the volume
9609:Functional integrals
9504:
9475:
9407:
9258:
9108:
9009:
8986:
8894:
8711:
8624:
8554:(sometimes called a
8435:
8403:
8333:
8231:
8128:
8069:
7989:
7971:Multiple integration
7839:
7714:
7617:
7507:
7445:
7339:
7248:
6954:
6820:
6731:
6621:
6470:
6209:
6203:Minkowski inequality
6154:Minkowski inequality
5948:
5630:
5505:
5309:
5197:
5166:is a subinterval of
5062:
4905:
4761:
4478:
4409:measurable functions
4298:
4249:
4075:
4004:
3502:
3399:
3326:
3302:measurable functions
3224:
3156:. This area is just
3026:Lebesgue integration
3018:Lebesgue integration
2908:
2884:
2880:with mesh less than
2852:
2826:
2800:
2659:
2478:
2365:, on the interval .
2283:
2104:
2029:
1855:
1792:
1676:
1651:real-valued function
1434:hyperbolic logarithm
1372:
1335:
1302:
1199:method of exhaustion
1106:indefinite integrals
1007:Nonstandard analysis
475:Lebesgue integration
345:Rules and identities
116:
86:{\displaystyle f(x)}
68:
13740:Continuous function
13693:Functional analysis
13385:Young's convolution
13324:Measurable function
13207:Pythagorean theorem
13197:Parseval's identity
13146:Integrable function
12857:Contour integration
12747:Kolmogorov integral
12584:Crowell, Benjamin,
12363:10.21105/joss.01073
12354:2018JOSS....3.1073R
12228:2015JPhCS.626a2015K
11871:, Springer-Verlag,
11796:, pp. 139–140.
11760:, pp. 522–524.
11748:, pp. 519–520.
11688:, pp. 111–114.
11316:, pp. 286−287.
11197:, pp. 628–629.
11173:, pp. 385–386.
11161:, pp. 536–537.
11149:, pp. 215–216.
11137:, pp. 516–517.
11125:, pp. 305–306.
11074:, pp. 284–285.
11062:, pp. 201–204.
10878:
10804:
10642:Gaussian quadrature
10606:, and so on, where
10368:tables of integrals
10324:Parseval's identity
10224:
10174:be the function of
10102:
9925:
9779:
9625:functional integral
9603:time-scale calculus
9553:exterior derivative
9057:This is known as a
8962:Contour integration
8683:gravitational field
8395:dimensional region
8268:
8248:
8057:over the rectangle
7915:
7856:
7784:
7731:
7634:
7524:
7280:
7211:continuous function
7138:
7101:
7055:
7018:
6975:
6911:
6874:
6837:
6748:
6678:
6638:
6487:
6459:Riemann-integrable
5942:Hölder's inequality
5851:Hölder's inequality
5766:
5715:
5653:
5569:
5527:
5251:
5214:
5116:
5079:
4959:
4922:
4799:
4411:on a measure space
4190:
4150:
4092:
4066:linear combinations
4027:
3940:unbounded variation
3908:, and developed by
3250:
3217:is then defined by
2300:
2065:0, 1/5, 2/5, ..., 1
1966:
1935:
1872:
1693:
1559:Historical notation
1241:, the area under a
1188:History of calculus
678:Cauchy condensation
480:Contour integration
206:Fundamental theorem
133:
14155:Euclidean distance
14121:Recommender system
14001:Similarity measure
13815:evaluation metrics
13772:Mathematics portal
13655:Lists of integrals
13506:Probability theory
13408:Plancherel theorem
13314:Integral transform
13261:Chebyshev distance
13238:
13187:Euclidean distance
13120:Minkowski distance
12970:Skorokhod integral
12907:Dirichlet integral
12894:Improper integrals
12837:Reduction formulas
12772:Regulated integral
12737:Hellinger integral
12620:Integration Theory
12087:Hildebrandt, T. H.
11968:Folland, Gerald B.
11928:Dahlquist, Germund
11090:Reliable Computing
10989:Lists of integrals
10969:Mathematics portal
10941:
10848:
10790:
10745:
10726:partial derivative
10562:Runge's phenomenon
10524:
10497:method of brackets
10443:Legendre functions
10320:Meijer G-functions
10277:
10210:
10125:
10088:
10049:
10029:
10009:
9980:
9948:
9911:
9861:
9837:rectilinear motion
9835:. For example, in
9809:
9765:
9710:
9690:
9637:probability theory
9621:space of functions
9599:Lebesgue integrals
9569:divergence theorem
9537:
9481:
9461:
9390:
9210:
9044:
8992:
8931:
8768:
8749:
8649:
8540:
8506:
8418:
8374:
8309:
8254:
8234:
8166:
8111:
8027:
8016:
7938:
7895:
7894:
7842:
7807:
7770:
7769:
7717:
7689:
7677:
7620:
7596:Improper integrals
7577:
7510:
7483:
7374:
7303:
7266:
7162:
7160:
7124:
7087:
7041:
7004:
6961:
6934:
6897:
6860:
6823:
6774:
6734:
6701:
6664:
6624:
6507:
6473:
6428:
6129:
5801:
5752:
5701:
5639:
5602:
5555:
5513:
5465:
5299:pointwise products
5274:
5237:
5200:
5139:
5102:
5065:
4982:
4945:
4908:
4843:
4785:
4757:, it follows that
4511:
4394:
4278:
4214:
4176:
4136:
4078:
4047:
4013:
3995:pointwise addition
3815:
3813:
3806:
3650:
3482:
3371:
3277:
3236:
3028:
2987:
2890:
2870:
2838:
2812:
2716:
2611:
2422:Formal definitions
2331:
2286:
2258:
2257:
2039:
2005:
1970:
1952:
1921:
1858:
1820:
1729:The integral sign
1716:
1679:
1575:, from the letter
1475:Leibniz and Newton
1378:
1358:
1321:
1071:, the other being
850:Partial derivative
779:generalized Stokes
673:Alternating series
554:Reduction formulae
543:Heaviside's method
524:tangent half-angle
511:Cylindrical shells
434:Integral transform
429:Lists of integrals
233:Mean value theorem
188:
119:
95:
83:
14181:
14180:
14150:Cosine similarity
13986:Hopkins statistic
13779:
13778:
13745:Special functions
13708:Harmonic analysis
13546:
13545:
13479:
13478:
13294:Almost everywhere
13079: &
13029:
13028:
12932:Frullani integral
12902:Gaussian integral
12852:Laplace transform
12827:Inverse functions
12817:Partial fractions
12742:Khinchin integral
12702:Lebesgue integral
12478:978-0-387-95452-3
12427:978-0-691-11880-2
12392:978-0-07-100276-9
12333:978-3-319-12481-0
12251:Krantz, Steven G.
12193:978-0-321-38700-4
12144:978-0-13-627258-8
12137:, Prentice Hall,
12077:978-0-486-42084-4
11920:978-0-486-67766-8
11897:978-0-07-338315-6
11865:Bourbaki, Nicolas
11857:978-0-471-00005-1
11834:978-1-118-88382-2
11676:, pp. 88–89.
11364:, pp. 57–58.
10977:Integral equation
10704:of an equivalent
10465:-finite functions
10439:special functions
10350:list of integrals
10343:shell integration
10328:Gaussian integral
10052:{\displaystyle B}
10032:{\displaystyle A}
9713:{\displaystyle r}
9087:differential form
9071:Differential form
8956:Contour integrals
8479:
8065:of two intervals
8063:Cartesian product
7977:Multiple integral
7879:
7754:
7702:Riemann integrals
7669:
7666:
7612:improper integral
7602:Improper integral
6594:, are called the
6451:In this section,
5825:square-integrable
4662:, is necessarily
4597:, generalized by
4438:topological field
4062:linear functional
3906:Jaroslav Kurzweil
3802:
3764:
3646:
3608:
3319:-axis is finite:
3012:Lebesgue integral
2465:tagged partitions
2326:
2306:
2241:
2228:
2213:
2212:
2187:
2174:
2159:
2158:
2133:
2118:
2117:
2037:
1381:{\displaystyle k}
1149:Lebesgue integral
1084:definite integral
1045:
1044:
925:
924:
887:
886:
855:Multiple integral
791:
790:
695:
694:
662:Direct comparison
633:Convergence tests
571:
570:
539:Partial fractions
406:
405:
316:Second derivative
16:(Redirected from
14211:
14173:Confusion matrix
13948:Logarithmic Loss
13813:Machine learning
13806:
13799:
13792:
13783:
13782:
13698:Fourier analysis
13678:Complex analysis
13579:Major topics in
13573:
13566:
13559:
13550:
13549:
13496:Fourier analysis
13454:Milman's reverse
13437:
13435:Lebesgue measure
13429:
13428:
13413:Riemann–Lebesgue
13256:Bounded function
13247:
13245:
13244:
13239:
13237:
13236:
13156:Taxicab geometry
13111:Measurable space
13056:
13049:
13042:
13033:
13032:
12877:Trapezoidal rule
12862:Laplace's method
12762:Pfeffer integral
12722:Darboux integral
12717:Daniell integral
12712:Bochner integral
12707:Burkill integral
12697:Riemann integral
12674:
12667:
12660:
12651:
12650:
12626:Sloughter, Dan,
12599:Hussain, Faraz,
12568:Stroyan, K. D.,
12541:
12515:
12503:
12487:Struik, Dirk Jan
12481:
12465:Bulirsch, Roland
12456:
12445:
12430:
12409:
12395:
12374:
12365:
12336:
12316:
12283:
12269:
12246:
12221:
12196:
12171:
12170:
12169:
12159:
12147:
12136:
12125:Kahaner, David;
12121:
12112:
12080:
12057:
12040:
12030:
12015:Open Mathematics
12009:
11998:
11984:
11963:
11961:
11947:
11942:, archived from
11938:, Philadelphia:
11923:
11900:
11881:
11860:
11837:
11809:
11803:
11797:
11791:
11785:
11779:
11773:
11767:
11761:
11755:
11749:
11743:
11737:
11731:
11725:
11719:
11713:
11707:
11701:
11695:
11689:
11683:
11677:
11671:
11665:
11659:
11653:
11647:
11641:
11635:
11629:
11623:
11617:
11611:
11605:
11599:
11593:
11587:
11581:
11575:
11569:
11563:
11557:
11551:
11542:
11536:
11530:
11524:
11518:
11512:
11506:
11500:
11494:
11488:
11482:
11476:
11470:
11464:
11455:
11449:
11443:
11437:
11431:
11425:
11416:
11410:
11401:
11395:
11389:
11383:
11377:
11376:, p. IV.43.
11371:
11365:
11359:
11353:
11347:
11341:
11335:
11329:
11323:
11317:
11311:
11305:
11299:
11293:
11287:
11281:
11275:
11266:
11260:
11254:
11248:
11242:
11232:
11226:
11216:
11210:
11204:
11198:
11192:
11186:
11180:
11174:
11168:
11162:
11156:
11150:
11144:
11138:
11132:
11126:
11120:
11114:
11113:
11081:
11075:
11069:
11063:
11057:
11051:
11045:
11039:
11033:
11016:
11005:
10971:
10966:
10965:
10950:
10948:
10947:
10942:
10934:
10933:
10909:
10908:
10877:
10866:
10855:
10854:
10803:
10798:
10754:
10752:
10751:
10746:
10744:
10743:
10659:
10651:
10639:
10628:
10617:
10605:
10591:
10576:Romberg's method
10557:
10552:
10536:trapezoidal rule
10532:rectangle method
10502:Mellin transform
10339:disk integration
10286:
10284:
10283:
10278:
10223:
10218:
10195:
10185:
10181:
10177:
10173:
10134:
10132:
10131:
10126:
10101:
10096:
10084:
10083:
10058:
10056:
10055:
10050:
10038:
10036:
10035:
10030:
10018:
10016:
10015:
10010:
9989:
9987:
9986:
9981:
9957:
9955:
9954:
9949:
9924:
9919:
9870:
9868:
9867:
9864:{\displaystyle }
9862:
9818:
9816:
9815:
9810:
9789:
9788:
9778:
9773:
9754:
9748:
9742:
9736:
9725:
9719:
9717:
9716:
9711:
9699:
9697:
9696:
9691:
9686:
9685:
9665:disc integration
9546:
9544:
9543:
9538:
9536:
9525:
9514:
9490:
9488:
9487:
9482:
9470:
9468:
9467:
9462:
9399:
9397:
9396:
9391:
9219:
9217:
9216:
9211:
9059:contour integral
9053:
9051:
9050:
9045:
9021:
9020:
9001:
8999:
8998:
8993:
8981:
8977:
8968:complex analysis
8950:electromagnetism
8946:classical theory
8940:
8938:
8937:
8932:
8927:
8926:
8913:
8912:
8906:
8905:
8886:
8876:
8866:
8856:
8852:
8841:
8835:
8824:
8818:
8812:
8806:
8772:surface integral
8758:
8756:
8755:
8750:
8745:
8734:
8729:
8728:
8703:
8690:
8676:
8667:
8658:
8656:
8655:
8650:
8645:
8637:
8616:
8610:
8568:contour integral
8530:Surface integral
8515:
8513:
8512:
8507:
8492:
8491:
8477:
8476:
8471:
8470:
8458:
8447:
8446:
8427:
8425:
8424:
8419:
8417:
8416:
8411:
8383:
8381:
8380:
8375:
8345:
8344:
8318:
8316:
8315:
8310:
8298:
8294:
8267:
8262:
8247:
8242:
8222:Fubini's theorem
8220:is continuous),
8212:over the domain
8211:
8184:
8175:
8173:
8172:
8167:
8140:
8139:
8120:
8118:
8117:
8112:
8025:
8023:
8022:
8017:
7966:
7962:
7947:
7945:
7944:
7939:
7914:
7909:
7893:
7855:
7850:
7831:
7816:
7814:
7813:
7808:
7783:
7778:
7768:
7730:
7725:
7686:
7684:
7683:
7678:
7670:
7668:
7667:
7662:
7644:
7636:
7633:
7628:
7586:
7584:
7583:
7578:
7523:
7518:
7499:
7492:
7490:
7489:
7484:
7470:
7437:
7433:
7429:
7425:
7414:
7402:
7390:
7383:
7381:
7380:
7375:
7349:
7331:
7319:
7312:
7310:
7309:
7304:
7279:
7274:
7240:
7236:
7228:
7186:
7182:
7178:
7171:
7169:
7168:
7163:
7161:
7137:
7132:
7100:
7095:
7083:
7078:
7054:
7049:
7017:
7012:
7000:
6974:
6969:
6943:
6941:
6940:
6935:
6910:
6905:
6873:
6868:
6836:
6831:
6806:
6802:
6798:
6783:
6781:
6780:
6775:
6747:
6742:
6724:, this implies:
6723:
6710:
6708:
6707:
6702:
6677:
6672:
6637:
6632:
6611:
6601:
6589:
6585:
6581:
6577:
6566:
6533:
6529:
6516:
6514:
6513:
6508:
6486:
6481:
6454:
6437:
6435:
6434:
6429:
6424:
6423:
6419:
6410:
6406:
6398:
6397:
6392:
6388:
6357:
6356:
6352:
6343:
6339:
6331:
6330:
6325:
6321:
6290:
6289:
6285:
6276:
6272:
6264:
6263:
6258:
6254:
6200:
6198:
6186:
6184:
6178:
6170:
6166:
6162:
6149:
6138:
6136:
6135:
6130:
6125:
6124:
6120:
6111:
6107:
6099:
6098:
6093:
6089:
6061:
6060:
6056:
6047:
6043:
6035:
6034:
6029:
6025:
5994:
5990:
5939:
5937:
5929:
5927:
5919:
5915:
5911:
5909:
5907:
5906:
5901:
5898:
5891:
5889:
5888:
5883:
5880:
5872:
5860:
5856:
5846:
5835:on the interval
5834:
5830:
5810:
5808:
5807:
5802:
5797:
5793:
5785:
5784:
5765:
5760:
5746:
5742:
5734:
5733:
5714:
5709:
5692:
5691:
5686:
5682:
5652:
5647:
5625:
5619:
5615:
5611:
5609:
5608:
5603:
5591:
5574:
5568:
5563:
5551:
5547:
5526:
5521:
5500:
5498:
5490:
5478:
5474:
5472:
5471:
5466:
5461:
5444:
5427:
5419:
5410:
5409:
5373:
5372:
5301:and powers, and
5296:
5292:
5283:
5281:
5280:
5275:
5250:
5245:
5213:
5208:
5192:
5188:
5177:
5165:
5148:
5146:
5145:
5140:
5115:
5110:
5078:
5073:
5057:
5045:
5041:
5022:
5010:
5006:
4991:
4989:
4988:
4983:
4958:
4953:
4921:
4916:
4900:
4896:
4892:
4880:
4876:
4852:
4850:
4849:
4844:
4798:
4793:
4756:
4741:
4726:
4714:
4710:
4698:
4694:
4676:
4672:
4661:
4649:
4638:
4603:Hildebrandt 1953
4599:Nicolas Bourbaki
4596:
4580:
4576:
4566:
4562:
4554:
4543:
4537:
4531:
4527:
4520:
4518:
4517:
4512:
4496:
4495:
4470:
4464:
4460:
4456:
4435:
4422:
4403:
4401:
4400:
4395:
4380:
4379:
4354:
4353:
4310:
4309:
4287:
4285:
4284:
4279:
4267:
4266:
4241:
4237:
4223:
4221:
4220:
4215:
4189:
4184:
4149:
4144:
4091:
4086:
4056:
4054:
4053:
4048:
4026:
4021:
3969:Bochner integral
3962:Choquet integral
3868:Daniell integral
3843:Riemann integral
3839:Darboux integral
3824:
3822:
3821:
3816:
3814:
3810:
3809:
3803:
3800:
3765:
3762:
3732:
3727:
3722:
3690:
3685:
3680:
3669:
3668:
3658:
3654:
3653:
3647:
3644:
3609:
3606:
3579:
3574:
3569:
3540:
3535:
3530:
3519:
3518:
3508:
3491:
3489:
3488:
3483:
3474:
3473:
3464:
3463:
3444:
3443:
3434:
3433:
3411:
3410:
3391:
3387:
3380:
3378:
3377:
3372:
3351:
3343:
3338:
3337:
3318:
3314:
3310:
3295:
3286:
3284:
3283:
3278:
3260:
3259:
3249:
3244:
3216:
3212:
3181:
3155:
3141:
3131:
3127:
3113:
3106:
3096:
3089:
3078:Lebesgue measure
3071:
3067:
3055:
3006:Darboux integral
2996:
2994:
2993:
2988:
2977:
2973:
2972:
2971:
2958:
2957:
2941:
2936:
2899:
2897:
2896:
2891:
2879:
2877:
2876:
2873:{\displaystyle }
2871:
2847:
2845:
2844:
2839:
2821:
2819:
2818:
2813:
2791:
2787:
2782:Riemann integral
2779:
2756:
2725:
2723:
2722:
2717:
2712:
2711:
2698:
2697:
2681:
2676:
2651:
2643:
2631:
2627:
2620:
2618:
2617:
2612:
2599:
2598:
2586:
2585:
2573:
2572:
2548:
2547:
2535:
2534:
2522:
2521:
2509:
2508:
2496:
2495:
2455:Riemann integral
2449:Riemann integral
2434:
2414:
2402:
2393:
2381:
2364:
2358:
2357:
2356:
2347:
2340:
2338:
2337:
2332:
2327:
2319:
2307:
2302:
2299:
2294:
2275:
2267:
2265:
2264:
2259:
2247:
2243:
2242:
2234:
2229:
2221:
2214:
2205:
2204:
2193:
2189:
2188:
2180:
2175:
2167:
2160:
2151:
2150:
2145:
2141:
2134:
2126:
2119:
2110:
2109:
2096:
2095:
2094:
2088:
2087:
2081:
2080:
2074:
2073:
2066:
2062:
2055:
2048:
2046:
2045:
2040:
2038:
2033:
2025:
2002:
2001:
2000:
1979:
1977:
1976:
1971:
1965:
1960:
1951:
1950:
1934:
1929:
1920:
1919:
1901:
1900:
1885:
1884:
1871:
1866:
1850:
1829:
1827:
1826:
1821:
1777:
1771:
1765:
1755:. The function
1754:
1748:
1743:of the variable
1738:
1732:
1725:
1723:
1722:
1717:
1692:
1687:
1668:
1662:
1628:
1621:
1620:
1619:
1618:
1613:
1583:), standing for
1553:hyperreal number
1529:Fourier analysis
1470:
1419:
1412:
1387:
1385:
1384:
1379:
1367:
1365:
1364:
1359:
1350:
1349:
1330:
1328:
1327:
1322:
1320:
1319:
1293:
1289:
1286:
1223:area of a circle
1209:and philosopher
1168:surface integral
1137:Bernhard Riemann
1037:
1030:
1023:
971:
936:
902:
901:
898:
865:Surface integral
808:
807:
804:
712:
711:
708:
668:Limit comparison
588:
587:
584:
470:Riemann integral
423:
422:
419:
379:L'Hôpital's rule
336:Taylor's theorem
257:
256:
253:
197:
195:
194:
189:
141:
132:
127:
97:
96:
92:
90:
89:
84:
21:
14219:
14218:
14214:
14213:
14212:
14210:
14209:
14208:
14184:
14183:
14182:
14177:
14164:
14138:
14115:
14106:Inception score
14094:
14071:
14049:Computer Vision
14043:
14015:
13952:
13884:
13816:
13810:
13780:
13775:
13764:
13713:P-adic analysis
13664:
13650:Matrix calculus
13645:Tensor calculus
13640:Vector calculus
13603:Differentiation
13583:
13577:
13547:
13542:
13475:
13432:
13427:
13389:
13365:Hausdorff–Young
13345:Babenko–Beckner
13333:
13282:
13232:
13228:
13226:
13223:
13222:
13216:
13160:
13129:
13125:Sequence spaces
13065:
13060:
13030:
13025:
13001:Integration Bee
12974:
12941:
12888:
12884:Risch algorithm
12822:Euler's formula
12782:
12776:
12757:Pettis integral
12689:
12683:
12678:
12617:Kowalk, W. P.,
12593:Garrett, Paul,
12558:
12526:
12523:
12518:
12506:
12501:
12479:
12454:
12436:Stillwell, John
12428:
12400:Saks, Stanisław
12393:
12334:
12314:
12267:
12194:
12176:Katz, Victor J.
12167:
12165:
12157:
12145:
12081:
12078:
11999:
11982:
11952:Feller, William
11921:
11905:Cajori, Florian
11898:
11879:
11858:
11842:Apostol, Tom M.
11835:
11818:
11813:
11812:
11804:
11800:
11792:
11788:
11780:
11776:
11768:
11764:
11756:
11752:
11744:
11740:
11732:
11728:
11720:
11716:
11708:
11704:
11696:
11692:
11684:
11680:
11672:
11668:
11660:
11656:
11648:
11644:
11640:, p. 1024.
11636:
11632:
11628:, p. 1014.
11624:
11620:
11612:
11608:
11600:
11596:
11588:
11584:
11576:
11572:
11564:
11560:
11552:
11545:
11537:
11533:
11525:
11521:
11513:
11509:
11501:
11497:
11489:
11485:
11477:
11473:
11465:
11458:
11450:
11446:
11438:
11434:
11426:
11419:
11411:
11404:
11396:
11392:
11384:
11380:
11372:
11368:
11360:
11356:
11348:
11344:
11336:
11332:
11324:
11320:
11312:
11308:
11300:
11296:
11288:
11284:
11276:
11269:
11261:
11257:
11249:
11245:
11233:
11229:
11221:, p. 414;
11217:
11213:
11205:
11201:
11193:
11189:
11181:
11177:
11169:
11165:
11157:
11153:
11145:
11141:
11133:
11129:
11121:
11117:
11082:
11078:
11070:
11066:
11058:
11054:
11046:
11042:
11034:
11030:
11025:
11020:
11019:
11006:
11002:
10997:
10983:Integral symbol
10967:
10960:
10957:
10929:
10928:
10904:
10903:
10867:
10856:
10850:
10849:
10799:
10794:
10788:
10785:
10784:
10774:
10769:
10739:
10735:
10733:
10730:
10729:
10718:differentiation
10714:
10695:
10689:
10673:
10653:
10649:
10634:
10627:
10619:
10616:
10607:
10603:
10593:
10589:
10579:
10555:
10550:
10516:
10510:
10415:Risch algorithm
10389:involving only
10364:
10358:
10219:
10214:
10208:
10205:
10204:
10187:
10183:
10179:
10175:
10164:
10157:
10152:
10097:
10092:
10073:
10069:
10067:
10064:
10063:
10044:
10041:
10040:
10024:
10021:
10020:
9995:
9992:
9991:
9966:
9963:
9962:
9920:
9915:
9879:
9876:
9875:
9844:
9841:
9840:
9784:
9780:
9774:
9769:
9760:
9757:
9756:
9750:
9744:
9738:
9727:
9721:
9705:
9702:
9701:
9681:
9677:
9672:
9669:
9668:
9641:random variable
9633:
9617:
9611:
9591:
9585:
9573:Green's theorem
9565:Stokes' theorem
9532:
9521:
9510:
9505:
9502:
9501:
9476:
9473:
9472:
9408:
9405:
9404:
9259:
9256:
9255:
9109:
9106:
9105:
9083:
9073:
9067:
9016:
9012:
9010:
9007:
9006:
8987:
8984:
8983:
8979:
8975:
8964:
8958:
8922:
8921:
8908:
8907:
8901:
8897:
8895:
8892:
8891:
8882:
8872:
8862:
8854:
8843:
8837:
8826:
8820:
8814:
8808:
8802:
8784:double integral
8741:
8730:
8724:
8720:
8712:
8709:
8708:
8692:
8686:
8672:
8663:
8641:
8633:
8625:
8622:
8621:
8612:
8606:
8532:
8524:Main articles:
8522:
8487:
8483:
8472:
8466:
8462:
8454:
8442:
8438:
8436:
8433:
8432:
8412:
8407:
8406:
8404:
8401:
8400:
8340:
8336:
8334:
8331:
8330:
8263:
8258:
8253:
8249:
8243:
8238:
8232:
8229:
8228:
8194:
8187:double integral
8180:
8135:
8131:
8129:
8126:
8125:
8121:can be written
8070:
8067:
8066:
8039:double integral
7990:
7987:
7986:
7979:
7973:
7964:
7960:
7910:
7899:
7883:
7851:
7846:
7840:
7837:
7836:
7821:
7779:
7774:
7758:
7726:
7721:
7715:
7712:
7711:
7661:
7645:
7637:
7635:
7629:
7624:
7618:
7615:
7614:
7604:
7598:
7593:
7519:
7514:
7508:
7505:
7504:
7497:
7463:
7446:
7443:
7442:
7435:
7431:
7427:
7423:
7419:that admits an
7417:closed interval
7412:
7409:
7392:
7388:
7342:
7340:
7337:
7336:
7321:
7317:
7275:
7270:
7249:
7246:
7245:
7238:
7234:
7231:closed interval
7226:
7223:
7207:differentiation
7199:
7193:
7184:
7180:
7176:
7159:
7158:
7133:
7128:
7096:
7091:
7082:
7076:
7075:
7050:
7045:
7013:
7008:
6999:
6995:
6970:
6965:
6957:
6955:
6952:
6951:
6906:
6901:
6869:
6864:
6832:
6827:
6821:
6818:
6817:
6804:
6800:
6796:
6743:
6738:
6732:
6729:
6728:
6715:
6673:
6668:
6633:
6628:
6622:
6619:
6618:
6603:
6599:
6587:
6583:
6579:
6576:
6568:
6561:
6552:
6545:
6535:
6531:
6521:
6482:
6477:
6471:
6468:
6467:
6463:. The integral
6452:
6449:
6415:
6411:
6393:
6375:
6371:
6370:
6366:
6362:
6361:
6348:
6344:
6326:
6308:
6304:
6303:
6299:
6295:
6294:
6281:
6277:
6259:
6226:
6222:
6221:
6217:
6213:
6212:
6210:
6207:
6206:
6190:
6188:
6180:
6174:
6172:
6168:
6164:
6157:
6156:. Suppose that
6140:
6116:
6112:
6094:
6076:
6072:
6071:
6067:
6063:
6062:
6052:
6048:
6030:
6012:
6008:
6007:
6003:
5999:
5998:
5955:
5951:
5949:
5946:
5945:
5933:
5931:
5923:
5921:
5917:
5913:
5902:
5899:
5896:
5895:
5893:
5884:
5881:
5878:
5877:
5875:
5874:
5862:
5858:
5854:
5853:. Suppose that
5836:
5832:
5828:
5780:
5776:
5761:
5756:
5751:
5747:
5729:
5725:
5710:
5705:
5700:
5696:
5687:
5648:
5643:
5638:
5634:
5633:
5631:
5628:
5627:
5621:
5617:
5613:
5587:
5570:
5564:
5559:
5522:
5517:
5512:
5508:
5506:
5503:
5502:
5494:
5492:
5480:
5476:
5457:
5440:
5423:
5415:
5405:
5401:
5368:
5364:
5310:
5307:
5306:
5303:absolute values
5294:
5290:
5246:
5241:
5209:
5204:
5198:
5195:
5194:
5190:
5179:
5167:
5155:
5111:
5106:
5074:
5069:
5063:
5060:
5059:
5047:
5043:
5024:
5012:
5008:
4993:
4954:
4949:
4917:
4912:
4906:
4903:
4902:
4898:
4894:
4882:
4878:
4859:
4794:
4789:
4762:
4759:
4758:
4743:
4728:
4716:
4712:
4700:
4696:
4678:
4674:
4670:
4651:
4647:
4628:
4611:
4594:
4578:
4568:
4564:
4560:
4553:
4545:
4539:
4533:
4529:
4525:
4491:
4487:
4479:
4476:
4475:
4466:
4462:
4458:
4440:
4433:
4425:locally compact
4412:
4375:
4371:
4349:
4345:
4305:
4301:
4299:
4296:
4295:
4262:
4258:
4250:
4247:
4246:
4239:
4235:
4185:
4180:
4145:
4140:
4087:
4082:
4076:
4073:
4072:
4022:
4017:
4005:
4002:
4001:
3987:
3982:
3951:semimartingales
3929:Brownian motion
3925:semimartingales
3831:
3829:Other integrals
3812:
3811:
3805:
3804:
3799:
3797:
3788:
3787:
3761:
3759:
3734:
3733:
3731:
3726:
3721:
3689:
3684:
3679:
3664:
3660:
3656:
3655:
3649:
3648:
3643:
3641:
3632:
3631:
3605:
3603:
3581:
3580:
3578:
3573:
3568:
3539:
3534:
3529:
3514:
3510:
3505:
3503:
3500:
3499:
3469:
3465:
3459:
3455:
3439:
3435:
3429:
3425:
3406:
3402:
3400:
3397:
3396:
3389:
3385:
3347:
3339:
3333:
3329:
3327:
3324:
3323:
3316:
3312:
3308:
3291:
3255:
3251:
3245:
3240:
3225:
3222:
3221:
3214:
3183:
3157:
3143:
3133:
3129:
3115:
3111:
3098:
3091:
3090:of an interval
3080:
3069:
3057:
3053:
3020:
3014:
2967:
2963:
2953:
2949:
2937:
2926:
2915:
2911:
2909:
2906:
2905:
2885:
2882:
2881:
2853:
2850:
2849:
2827:
2824:
2823:
2801:
2798:
2797:
2789:
2785:
2778:
2772:
2762:
2755:
2745:
2736:
2730:
2707:
2703:
2693:
2689:
2677:
2666:
2660:
2657:
2656:
2649:
2641:
2633:
2629:
2625:
2594:
2590:
2581:
2577:
2562:
2558:
2543:
2539:
2530:
2526:
2517:
2513:
2504:
2500:
2491:
2487:
2479:
2476:
2475:
2469:closed interval
2457:
2451:
2442:
2441:
2440:
2439:
2438:
2435:
2424:
2419:
2418:
2417:
2416:
2415:
2406:
2403:
2395:
2394:
2385:
2382:
2373:
2372:
2360:
2352:
2350:
2349:
2345:
2318:
2301:
2295:
2290:
2284:
2281:
2280:
2273:
2233:
2220:
2219:
2215:
2203:
2179:
2166:
2165:
2161:
2149:
2125:
2124:
2120:
2108:
2105:
2102:
2101:
2092:
2090:
2085:
2083:
2078:
2076:
2071:
2069:
2068:
2064:
2057:
2050:
2032:
2030:
2027:
2026:
2016:
1996:
1994:
1993:
1986:
1984:Interpretations
1961:
1956:
1946:
1942:
1930:
1925:
1915:
1911:
1896:
1892:
1880:
1876:
1867:
1862:
1856:
1853:
1852:
1846:
1793:
1790:
1789:
1773:
1767:
1756:
1750:
1744:
1734:
1730:
1688:
1683:
1677:
1674:
1673:
1664:
1653:
1647:
1639:Jacob Bernoulli
1635:
1623:
1614:
1609:
1608:
1607:
1606:
1569:integral symbol
1561:
1543:(a subfield of
1509:Bishop Berkeley
1501:
1477:
1468:
1457:differentiation
1414:
1408:
1373:
1370:
1369:
1345:
1341:
1336:
1333:
1332:
1315:
1311:
1303:
1300:
1299:
1291:
1287:
1195:
1190:
1184:
1073:differentiation
1041:
1012:
1011:
997:Integration Bee
972:
969:
962:
961:
937:
934:
927:
926:
899:
896:
889:
888:
870:Volume integral
805:
800:
793:
792:
709:
704:
697:
696:
666:
585:
580:
573:
572:
564:Risch algorithm
534:Euler's formula
420:
415:
408:
407:
389:General Leibniz
272:generalizations
254:
249:
242:
228:Rolle's theorem
223:
198:
134:
128:
123:
117:
114:
113:
69:
66:
65:
54:
43:
28:
23:
22:
15:
12:
11:
5:
14217:
14207:
14206:
14201:
14196:
14179:
14178:
14176:
14175:
14169:
14166:
14165:
14163:
14162:
14157:
14152:
14146:
14144:
14140:
14139:
14137:
14136:
14131:
14125:
14123:
14117:
14116:
14114:
14113:
14108:
14102:
14100:
14096:
14095:
14093:
14092:
14087:
14081:
14079:
14073:
14072:
14070:
14069:
14064:
14059:
14053:
14051:
14045:
14044:
14042:
14041:
14036:
14031:
14025:
14023:
14017:
14016:
14014:
14013:
14008:
14003:
13998:
13993:
13988:
13983:
13978:
13976:Davies-Bouldin
13973:
13968:
13962:
13960:
13954:
13953:
13951:
13950:
13945:
13940:
13935:
13930:
13925:
13920:
13915:
13910:
13905:
13900:
13894:
13892:
13890:Classification
13886:
13885:
13883:
13882:
13877:
13872:
13867:
13862:
13857:
13852:
13847:
13842:
13837:
13832:
13826:
13824:
13818:
13817:
13809:
13808:
13801:
13794:
13786:
13777:
13776:
13769:
13766:
13765:
13763:
13762:
13757:
13752:
13747:
13742:
13737:
13731:
13730:
13725:
13723:Measure theory
13720:
13717:P-adic numbers
13710:
13705:
13700:
13695:
13690:
13680:
13675:
13669:
13666:
13665:
13663:
13662:
13657:
13652:
13647:
13642:
13637:
13632:
13627:
13626:
13625:
13620:
13615:
13605:
13600:
13588:
13585:
13584:
13576:
13575:
13568:
13561:
13553:
13544:
13543:
13541:
13540:
13539:
13538:
13533:
13523:
13518:
13513:
13508:
13503:
13498:
13493:
13487:
13485:
13481:
13480:
13477:
13476:
13474:
13473:
13468:
13463:
13458:
13457:
13456:
13446:
13440:
13438:
13426:
13425:
13420:
13415:
13410:
13405:
13399:
13397:
13391:
13390:
13388:
13387:
13382:
13377:
13372:
13367:
13362:
13357:
13352:
13347:
13341:
13339:
13335:
13334:
13332:
13331:
13326:
13321:
13316:
13311:
13309:Function space
13306:
13301:
13296:
13290:
13288:
13284:
13283:
13281:
13280:
13275:
13274:
13273:
13263:
13258:
13252:
13250:
13235:
13231:
13218:
13217:
13215:
13214:
13209:
13204:
13199:
13194:
13189:
13184:
13182:Cauchy–Schwarz
13179:
13173:
13171:
13162:
13161:
13159:
13158:
13153:
13148:
13142:
13140:
13131:
13130:
13128:
13127:
13122:
13117:
13108:
13103:
13102:
13101:
13091:
13083:
13081:Hilbert spaces
13073:
13071:
13070:Basic concepts
13067:
13066:
13059:
13058:
13051:
13044:
13036:
13027:
13026:
13024:
13023:
13022:
13021:
13016:
13008:
13003:
12998:
12996:Gabriel's horn
12993:
12988:
12982:
12980:
12976:
12975:
12973:
12972:
12967:
12962:
12957:
12951:
12949:
12943:
12942:
12940:
12939:
12934:
12929:
12924:
12923:
12922:
12917:
12909:
12904:
12898:
12896:
12890:
12889:
12887:
12886:
12881:
12880:
12879:
12874:
12872:Simpson's rule
12864:
12859:
12854:
12849:
12844:
12839:
12834:
12832:Changing order
12829:
12824:
12819:
12814:
12809:
12808:
12807:
12802:
12797:
12786:
12784:
12778:
12777:
12775:
12774:
12769:
12764:
12759:
12754:
12749:
12744:
12739:
12734:
12729:
12724:
12719:
12714:
12709:
12704:
12699:
12693:
12691:
12685:
12684:
12677:
12676:
12669:
12662:
12654:
12648:
12647:
12640:
12631:
12624:
12615:
12604:
12597:
12591:
12582:
12573:
12566:
12557:
12554:
12553:
12552:
12542:
12522:
12521:External links
12519:
12517:
12516:
12504:
12499:
12489:, ed. (1986),
12483:
12477:
12457:
12452:
12432:
12426:
12410:
12396:
12391:
12375:
12337:
12332:
12317:
12313:978-0821827833
12312:
12284:
12270:
12265:
12247:
12210:IOP Publishing
12197:
12192:
12184:Addison-Wesley
12172:
12148:
12143:
12122:
12103:(2): 111–139,
12083:
12076:
12064:, ed. (2002),
12058:
12021:(1): 983–995,
12010:
11985:
11980:
11964:
11948:
11924:
11919:
11901:
11896:
11883:
11877:
11861:
11856:
11838:
11833:
11819:
11817:
11814:
11811:
11810:
11798:
11786:
11784:, p. 147.
11774:
11772:, p. 144.
11762:
11750:
11738:
11726:
11714:
11712:, p. 116.
11702:
11700:, p. 306.
11690:
11678:
11666:
11654:
11642:
11630:
11618:
11616:, p. 991.
11606:
11604:, p. 697.
11594:
11592:, p. 981.
11582:
11580:, p. 980.
11570:
11568:, p. 897.
11558:
11556:, p. 896.
11543:
11541:, p. 895.
11531:
11529:, p. 418.
11519:
11517:, p. 416.
11507:
11505:, p. 355.
11495:
11493:, p. 205.
11483:
11481:, p. 202.
11471:
11456:
11444:
11432:
11417:
11402:
11390:
11378:
11366:
11354:
11352:, p. 796.
11342:
11330:
11328:, p. 173.
11318:
11306:
11294:
11292:, p. 259.
11282:
11267:
11265:, p. 182.
11255:
11253:, p. 246.
11243:
11227:
11225:, p. 154.
11211:
11209:, p. 785.
11199:
11187:
11185:, p. 131.
11183:Stillwell 1989
11175:
11163:
11151:
11139:
11127:
11115:
11096:(2): 191–197.
11076:
11064:
11052:
11040:
11038:, p. 117.
11027:
11026:
11024:
11021:
11018:
11017:
11015:, for example.
10999:
10998:
10996:
10993:
10992:
10991:
10986:
10980:
10973:
10972:
10956:
10953:
10952:
10951:
10940:
10937:
10932:
10927:
10924:
10921:
10918:
10915:
10912:
10907:
10902:
10899:
10896:
10893:
10890:
10887:
10884:
10881:
10876:
10873:
10870:
10865:
10862:
10859:
10853:
10847:
10844:
10841:
10838:
10835:
10832:
10829:
10826:
10823:
10819:
10816:
10813:
10810:
10807:
10802:
10797:
10793:
10773:
10770:
10768:
10765:
10742:
10738:
10713:
10710:
10691:Main article:
10688:
10685:
10672:
10669:
10623:
10611:
10601:
10587:
10540:Simpson's rule
10512:Main article:
10509:
10506:
10451:gamma function
10360:Main article:
10357:
10354:
10288:
10287:
10276:
10273:
10270:
10267:
10264:
10261:
10258:
10255:
10252:
10249:
10246:
10243:
10240:
10236:
10233:
10230:
10227:
10222:
10217:
10213:
10156:
10153:
10151:
10148:
10140:thermodynamics
10136:
10135:
10124:
10121:
10118:
10114:
10111:
10108:
10105:
10100:
10095:
10091:
10087:
10082:
10079:
10076:
10072:
10048:
10028:
10008:
10005:
10002:
9999:
9979:
9976:
9973:
9970:
9959:
9958:
9947:
9944:
9941:
9937:
9934:
9931:
9928:
9923:
9918:
9914:
9910:
9907:
9904:
9901:
9898:
9895:
9892:
9889:
9886:
9883:
9860:
9857:
9854:
9851:
9848:
9808:
9805:
9802:
9798:
9795:
9792:
9787:
9783:
9777:
9772:
9768:
9764:
9709:
9689:
9684:
9680:
9676:
9632:
9629:
9613:Main article:
9610:
9607:
9587:Main article:
9584:
9581:
9535:
9531:
9528:
9524:
9520:
9517:
9513:
9509:
9480:
9460:
9457:
9454:
9451:
9448:
9445:
9442:
9439:
9436:
9433:
9430:
9427:
9424:
9421:
9418:
9415:
9412:
9401:
9400:
9389:
9386:
9383:
9380:
9377:
9374:
9370:
9367:
9364:
9361:
9358:
9355:
9352:
9349:
9346:
9343:
9340:
9337:
9334:
9331:
9327:
9324:
9321:
9318:
9315:
9312:
9309:
9306:
9303:
9300:
9297:
9294:
9291:
9288:
9284:
9281:
9278:
9275:
9272:
9269:
9266:
9263:
9221:
9220:
9209:
9206:
9202:
9199:
9196:
9193:
9190:
9187:
9184:
9181:
9178:
9175:
9172:
9168:
9165:
9162:
9159:
9156:
9153:
9150:
9147:
9144:
9141:
9138:
9134:
9131:
9128:
9125:
9122:
9119:
9116:
9113:
9069:Main article:
9066:
9063:
9055:
9054:
9043:
9040:
9037:
9033:
9030:
9027:
9024:
9019:
9015:
8991:
8960:Main article:
8957:
8954:
8942:
8941:
8930:
8925:
8920:
8916:
8911:
8904:
8900:
8879:surface normal
8877:with the unit
8786:analog of the
8760:
8759:
8748:
8744:
8740:
8737:
8733:
8727:
8723:
8719:
8716:
8679:electric field
8660:
8659:
8648:
8644:
8640:
8636:
8632:
8629:
8587:scalar product
8521:
8518:
8517:
8516:
8505:
8502:
8499:
8495:
8490:
8486:
8482:
8475:
8469:
8465:
8461:
8457:
8453:
8450:
8445:
8441:
8415:
8410:
8385:
8384:
8373:
8370:
8367:
8363:
8360:
8357:
8354:
8351:
8348:
8343:
8339:
8320:
8319:
8308:
8305:
8302:
8297:
8293:
8290:
8286:
8283:
8280:
8277:
8274:
8271:
8266:
8261:
8257:
8252:
8246:
8241:
8237:
8177:
8176:
8165:
8162:
8158:
8155:
8152:
8149:
8146:
8143:
8138:
8134:
8110:
8107:
8104:
8101:
8098:
8095:
8092:
8089:
8086:
8083:
8080:
8077:
8074:
8015:
8012:
8009:
8006:
8003:
8000:
7997:
7994:
7975:Main article:
7972:
7969:
7949:
7948:
7937:
7934:
7931:
7927:
7924:
7921:
7918:
7913:
7908:
7905:
7902:
7898:
7892:
7889:
7886:
7882:
7878:
7875:
7872:
7868:
7865:
7862:
7859:
7854:
7849:
7845:
7818:
7817:
7806:
7803:
7800:
7796:
7793:
7790:
7787:
7782:
7777:
7773:
7767:
7764:
7761:
7757:
7753:
7750:
7747:
7743:
7740:
7737:
7734:
7729:
7724:
7720:
7676:
7673:
7665:
7660:
7657:
7654:
7651:
7648:
7643:
7640:
7632:
7627:
7623:
7600:Main article:
7597:
7594:
7592:
7589:
7588:
7587:
7576:
7573:
7570:
7567:
7564:
7561:
7558:
7555:
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7546:
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7540:
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7527:
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7517:
7513:
7494:
7493:
7482:
7479:
7476:
7473:
7469:
7466:
7462:
7459:
7456:
7453:
7450:
7426:on . That is,
7421:antiderivative
7408:
7407:Second theorem
7405:
7385:
7384:
7373:
7370:
7367:
7364:
7361:
7358:
7355:
7352:
7348:
7345:
7314:
7313:
7302:
7299:
7296:
7292:
7289:
7286:
7283:
7278:
7273:
7269:
7265:
7262:
7259:
7256:
7253:
7222:
7219:
7195:Main article:
7192:
7189:
7173:
7172:
7157:
7154:
7150:
7147:
7144:
7141:
7136:
7131:
7127:
7123:
7120:
7117:
7113:
7110:
7107:
7104:
7099:
7094:
7090:
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7064:
7061:
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7053:
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7027:
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7021:
7016:
7011:
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6998:
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6987:
6984:
6981:
6978:
6973:
6968:
6964:
6960:
6959:
6945:
6944:
6933:
6930:
6927:
6923:
6920:
6917:
6914:
6909:
6904:
6900:
6896:
6893:
6890:
6886:
6883:
6880:
6877:
6872:
6867:
6863:
6859:
6856:
6853:
6849:
6846:
6843:
6840:
6835:
6830:
6826:
6785:
6784:
6773:
6770:
6767:
6764:
6760:
6757:
6754:
6751:
6746:
6741:
6737:
6712:
6711:
6700:
6697:
6694:
6690:
6687:
6684:
6681:
6676:
6671:
6667:
6663:
6660:
6657:
6654:
6650:
6647:
6644:
6641:
6636:
6631:
6627:
6572:
6557:
6550:
6543:
6518:
6517:
6506:
6503:
6499:
6496:
6493:
6490:
6485:
6480:
6476:
6448:
6445:
6444:
6443:
6427:
6422:
6418:
6414:
6409:
6405:
6402:
6396:
6391:
6387:
6384:
6381:
6378:
6374:
6369:
6365:
6360:
6355:
6351:
6347:
6342:
6338:
6335:
6329:
6324:
6320:
6317:
6314:
6311:
6307:
6302:
6298:
6293:
6288:
6284:
6280:
6275:
6271:
6268:
6262:
6257:
6253:
6250:
6247:
6244:
6241:
6238:
6235:
6232:
6229:
6225:
6220:
6216:
6179:|, |
6151:
6128:
6123:
6119:
6115:
6110:
6106:
6103:
6097:
6092:
6088:
6085:
6082:
6079:
6075:
6070:
6066:
6059:
6055:
6051:
6046:
6042:
6039:
6033:
6028:
6024:
6021:
6018:
6015:
6011:
6006:
6002:
5997:
5993:
5989:
5986:
5982:
5979:
5976:
5973:
5970:
5967:
5964:
5961:
5958:
5954:
5848:
5800:
5796:
5792:
5789:
5783:
5779:
5775:
5772:
5769:
5764:
5759:
5755:
5750:
5745:
5741:
5738:
5732:
5728:
5724:
5721:
5718:
5713:
5708:
5704:
5699:
5695:
5690:
5685:
5681:
5678:
5674:
5671:
5668:
5665:
5662:
5659:
5656:
5651:
5646:
5642:
5637:
5601:
5598:
5595:
5590:
5586:
5583:
5580:
5577:
5573:
5567:
5562:
5558:
5554:
5550:
5546:
5543:
5539:
5536:
5533:
5530:
5525:
5520:
5516:
5511:
5464:
5460:
5456:
5453:
5450:
5447:
5443:
5439:
5436:
5433:
5430:
5426:
5422:
5418:
5413:
5408:
5404:
5400:
5397:
5394:
5391:
5388:
5385:
5382:
5379:
5376:
5371:
5367:
5362:
5359:
5356:
5353:
5350:
5347:
5344:
5341:
5338:
5335:
5332:
5329:
5326:
5323:
5320:
5317:
5314:
5284:
5273:
5270:
5267:
5263:
5260:
5257:
5254:
5249:
5244:
5240:
5236:
5233:
5230:
5226:
5223:
5220:
5217:
5212:
5207:
5203:
5149:
5138:
5135:
5132:
5128:
5125:
5122:
5119:
5114:
5109:
5105:
5101:
5098:
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5088:
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4853:
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4830:
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4818:
4815:
4811:
4808:
4805:
4802:
4797:
4792:
4788:
4784:
4781:
4778:
4775:
4772:
4769:
4766:
4610:
4607:
4557:p-adic numbers
4549:
4522:
4521:
4509:
4506:
4503:
4499:
4494:
4490:
4486:
4483:
4465:or the symbol
4461:an element of
4405:
4404:
4393:
4390:
4387:
4383:
4378:
4374:
4370:
4367:
4364:
4361:
4357:
4352:
4348:
4344:
4341:
4338:
4335:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4308:
4304:
4289:
4288:
4277:
4274:
4270:
4265:
4261:
4257:
4254:
4225:
4224:
4212:
4209:
4206:
4202:
4199:
4196:
4193:
4188:
4183:
4179:
4175:
4172:
4169:
4166:
4162:
4159:
4156:
4153:
4148:
4143:
4139:
4135:
4132:
4129:
4126:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4090:
4085:
4081:
4058:
4057:
4046:
4043:
4039:
4036:
4033:
4030:
4025:
4020:
4016:
4012:
4009:
3986:
3983:
3981:
3978:
3977:
3976:
3965:
3958:
3943:
3936:Young integral
3932:
3913:
3910:Ralph Henstock
3890:
3879:
3864:
3853:
3846:
3830:
3827:
3826:
3825:
3808:
3798:
3796:
3793:
3790:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3760:
3758:
3755:
3752:
3749:
3746:
3743:
3740:
3739:
3737:
3730:
3725:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3688:
3683:
3681:
3678:
3675:
3672:
3667:
3663:
3659:
3657:
3652:
3642:
3640:
3637:
3634:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3604:
3602:
3599:
3596:
3593:
3590:
3587:
3586:
3584:
3577:
3572:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3546:
3543:
3538:
3533:
3531:
3528:
3525:
3522:
3517:
3513:
3509:
3507:
3493:
3492:
3481:
3478:
3472:
3468:
3462:
3458:
3454:
3451:
3448:
3442:
3438:
3432:
3428:
3424:
3421:
3418:
3414:
3409:
3405:
3382:
3381:
3370:
3367:
3364:
3361:
3358:
3355:
3350:
3346:
3342:
3336:
3332:
3288:
3287:
3276:
3273:
3269:
3266:
3263:
3258:
3254:
3248:
3243:
3239:
3235:
3232:
3229:
3097:is its width,
3037:Henri Lebesgue
3016:Main article:
3013:
3010:
2998:
2997:
2986:
2983:
2980:
2976:
2970:
2966:
2961:
2956:
2952:
2948:
2945:
2940:
2935:
2932:
2929:
2925:
2921:
2918:
2914:
2902:
2901:
2889:
2869:
2866:
2863:
2860:
2857:
2837:
2834:
2831:
2811:
2808:
2805:
2784:of a function
2774:
2764:
2750:
2741:
2732:
2727:
2726:
2715:
2710:
2706:
2701:
2696:
2692:
2688:
2685:
2680:
2675:
2672:
2669:
2665:
2648:of a function
2637:
2622:
2621:
2608:
2605:
2602:
2597:
2593:
2589:
2584:
2580:
2576:
2571:
2568:
2565:
2561:
2557:
2554:
2551:
2546:
2542:
2538:
2533:
2529:
2525:
2520:
2516:
2512:
2507:
2503:
2499:
2494:
2490:
2486:
2483:
2453:Main article:
2450:
2447:
2436:
2429:
2428:
2427:
2426:
2425:
2423:
2420:
2404:
2397:
2396:
2383:
2376:
2375:
2374:
2370:
2369:
2368:
2367:
2342:
2341:
2330:
2325:
2322:
2317:
2314:
2311:
2305:
2298:
2293:
2289:
2276:). One writes
2269:
2268:
2256:
2253:
2250:
2246:
2240:
2237:
2232:
2227:
2224:
2218:
2211:
2208:
2202:
2199:
2196:
2192:
2186:
2183:
2178:
2173:
2170:
2164:
2157:
2154:
2148:
2144:
2140:
2137:
2132:
2129:
2123:
2116:
2113:
2036:
1985:
1982:
1969:
1964:
1959:
1955:
1949:
1945:
1941:
1938:
1933:
1928:
1924:
1918:
1914:
1910:
1907:
1904:
1899:
1895:
1891:
1888:
1883:
1879:
1875:
1870:
1865:
1861:
1835:antiderivative
1831:
1830:
1819:
1816:
1813:
1809:
1806:
1803:
1800:
1797:
1781:
1727:
1726:
1715:
1712:
1709:
1705:
1702:
1699:
1696:
1691:
1686:
1682:
1646:
1643:
1634:
1631:
1593:Joseph Fourier
1560:
1557:
1541:measure theory
1500:
1497:
1476:
1473:
1436:, achieved by
1403:, and work by
1377:
1357:
1354:
1348:
1344:
1340:
1318:
1314:
1310:
1307:
1194:
1191:
1183:
1180:
1145:Henri Lebesgue
1101:antiderivative
1043:
1042:
1040:
1039:
1032:
1025:
1017:
1014:
1013:
1010:
1009:
1004:
999:
994:
992:List of topics
989:
984:
979:
973:
968:
967:
964:
963:
960:
959:
954:
949:
944:
938:
933:
932:
929:
928:
923:
922:
921:
920:
915:
910:
900:
895:
894:
891:
890:
885:
884:
883:
882:
877:
872:
867:
862:
857:
852:
844:
843:
839:
838:
837:
836:
831:
826:
821:
813:
812:
806:
799:
798:
795:
794:
789:
788:
787:
786:
781:
776:
771:
766:
761:
753:
752:
748:
747:
746:
745:
740:
735:
730:
725:
720:
710:
703:
702:
699:
698:
693:
692:
691:
690:
685:
680:
675:
670:
664:
659:
654:
649:
644:
636:
635:
629:
628:
627:
626:
621:
616:
611:
606:
601:
586:
579:
578:
575:
574:
569:
568:
567:
566:
561:
556:
551:
549:Changing order
546:
536:
531:
513:
508:
503:
495:
494:
493:Integration by
490:
489:
488:
487:
482:
477:
472:
467:
457:
455:Antiderivative
449:
448:
444:
443:
442:
441:
436:
431:
421:
414:
413:
410:
409:
404:
403:
402:
401:
396:
391:
386:
381:
376:
371:
366:
361:
356:
348:
347:
341:
340:
339:
338:
333:
328:
323:
318:
313:
305:
304:
300:
299:
298:
297:
296:
295:
290:
285:
275:
262:
261:
255:
248:
247:
244:
243:
241:
240:
235:
230:
224:
222:
221:
216:
210:
209:
208:
200:
199:
187:
184:
181:
178:
175:
172:
169:
166:
163:
160:
157:
154:
150:
147:
144:
140:
137:
131:
126:
122:
112:
109:
108:
102:
101:
82:
79:
76:
73:
32:antiderivative
26:
9:
6:
4:
3:
2:
14216:
14205:
14202:
14200:
14197:
14195:
14192:
14191:
14189:
14174:
14171:
14170:
14167:
14161:
14158:
14156:
14153:
14151:
14148:
14147:
14145:
14141:
14135:
14132:
14130:
14127:
14126:
14124:
14122:
14118:
14112:
14109:
14107:
14104:
14103:
14101:
14097:
14091:
14088:
14086:
14083:
14082:
14080:
14078:
14074:
14068:
14065:
14063:
14060:
14058:
14055:
14054:
14052:
14050:
14046:
14040:
14037:
14035:
14032:
14030:
14027:
14026:
14024:
14022:
14018:
14012:
14009:
14007:
14004:
14002:
13999:
13997:
13994:
13992:
13991:Jaccard index
13989:
13987:
13984:
13982:
13979:
13977:
13974:
13972:
13969:
13967:
13964:
13963:
13961:
13959:
13955:
13949:
13946:
13944:
13941:
13939:
13936:
13934:
13931:
13929:
13926:
13924:
13921:
13919:
13916:
13914:
13911:
13909:
13906:
13904:
13901:
13899:
13896:
13895:
13893:
13891:
13887:
13881:
13878:
13876:
13873:
13871:
13868:
13866:
13863:
13861:
13858:
13856:
13853:
13851:
13848:
13846:
13843:
13841:
13838:
13836:
13833:
13831:
13828:
13827:
13825:
13823:
13819:
13814:
13807:
13802:
13800:
13795:
13793:
13788:
13787:
13784:
13774:
13773:
13767:
13761:
13758:
13756:
13753:
13751:
13748:
13746:
13743:
13741:
13738:
13736:
13733:
13732:
13729:
13726:
13724:
13721:
13718:
13714:
13711:
13709:
13706:
13704:
13701:
13699:
13696:
13694:
13691:
13688:
13684:
13681:
13679:
13676:
13674:
13673:Real analysis
13671:
13670:
13667:
13661:
13658:
13656:
13653:
13651:
13648:
13646:
13643:
13641:
13638:
13636:
13633:
13631:
13628:
13624:
13621:
13619:
13616:
13614:
13611:
13610:
13609:
13606:
13604:
13601:
13599:
13595:
13594:
13590:
13589:
13586:
13582:
13574:
13569:
13567:
13562:
13560:
13555:
13554:
13551:
13537:
13534:
13532:
13529:
13528:
13527:
13524:
13522:
13521:Sobolev space
13519:
13517:
13516:Real analysis
13514:
13512:
13509:
13507:
13504:
13502:
13501:Lorentz space
13499:
13497:
13494:
13492:
13491:Bochner space
13489:
13488:
13486:
13482:
13472:
13469:
13467:
13464:
13462:
13459:
13455:
13452:
13451:
13450:
13447:
13445:
13442:
13441:
13439:
13436:
13430:
13424:
13421:
13419:
13416:
13414:
13411:
13409:
13406:
13404:
13401:
13400:
13398:
13396:
13392:
13386:
13383:
13381:
13378:
13376:
13373:
13371:
13368:
13366:
13363:
13361:
13358:
13356:
13353:
13351:
13348:
13346:
13343:
13342:
13340:
13336:
13330:
13327:
13325:
13322:
13320:
13317:
13315:
13312:
13310:
13307:
13305:
13302:
13300:
13297:
13295:
13292:
13291:
13289:
13285:
13279:
13276:
13272:
13269:
13268:
13267:
13264:
13262:
13259:
13257:
13254:
13253:
13251:
13249:
13229:
13219:
13213:
13210:
13208:
13205:
13203:
13200:
13198:
13195:
13193:
13192:Hilbert space
13190:
13188:
13185:
13183:
13180:
13178:
13175:
13174:
13172:
13170:
13168:
13163:
13157:
13154:
13152:
13149:
13147:
13144:
13143:
13141:
13139:
13137:
13132:
13126:
13123:
13121:
13118:
13116:
13112:
13109:
13107:
13106:Measure space
13104:
13100:
13097:
13096:
13095:
13092:
13090:
13088:
13084:
13082:
13078:
13075:
13074:
13072:
13068:
13064:
13057:
13052:
13050:
13045:
13043:
13038:
13037:
13034:
13020:
13017:
13015:
13012:
13011:
13009:
13007:
13004:
13002:
12999:
12997:
12994:
12992:
12989:
12987:
12986:Basel problem
12984:
12983:
12981:
12979:Miscellaneous
12977:
12971:
12968:
12966:
12963:
12961:
12958:
12956:
12953:
12952:
12950:
12948:
12944:
12938:
12935:
12933:
12930:
12928:
12925:
12921:
12918:
12916:
12913:
12912:
12910:
12908:
12905:
12903:
12900:
12899:
12897:
12895:
12891:
12885:
12882:
12878:
12875:
12873:
12870:
12869:
12868:
12865:
12863:
12860:
12858:
12855:
12853:
12850:
12848:
12845:
12843:
12840:
12838:
12835:
12833:
12830:
12828:
12825:
12823:
12820:
12818:
12815:
12813:
12810:
12806:
12803:
12801:
12798:
12796:
12795:Trigonometric
12793:
12792:
12791:
12788:
12787:
12785:
12779:
12773:
12770:
12768:
12765:
12763:
12760:
12758:
12755:
12753:
12750:
12748:
12745:
12743:
12740:
12738:
12735:
12733:
12732:Haar integral
12730:
12728:
12725:
12723:
12720:
12718:
12715:
12713:
12710:
12708:
12705:
12703:
12700:
12698:
12695:
12694:
12692:
12686:
12682:
12675:
12670:
12668:
12663:
12661:
12656:
12655:
12652:
12645:
12641:
12639:
12635:
12632:
12629:
12625:
12622:
12621:
12616:
12613:
12609:
12605:
12602:
12598:
12596:
12592:
12589:
12588:
12583:
12580:
12579:
12575:Mauch, Sean,
12574:
12571:
12567:
12564:
12560:
12559:
12550:
12549:Wolfram Alpha
12546:
12543:
12539:
12535:
12534:
12529:
12525:
12524:
12513:
12509:
12505:
12502:
12500:0-691-08404-1
12496:
12492:
12488:
12484:
12480:
12474:
12470:
12466:
12462:
12458:
12455:
12453:0-387-96981-0
12449:
12444:
12443:
12437:
12433:
12429:
12423:
12419:
12415:
12411:
12407:
12406:
12401:
12397:
12394:
12388:
12384:
12380:
12379:Rudin, Walter
12376:
12373:
12369:
12364:
12359:
12355:
12351:
12347:
12343:
12338:
12335:
12329:
12325:
12324:
12318:
12315:
12309:
12305:
12301:
12297:
12293:
12292:Loss, Michael
12289:
12288:Lieb, Elliott
12285:
12281:
12280:
12275:
12271:
12268:
12266:0-8493-7156-2
12262:
12259:, CRC Press,
12258:
12257:
12252:
12248:
12245:
12241:
12237:
12233:
12229:
12225:
12220:
12215:
12211:
12207:
12203:
12198:
12195:
12189:
12185:
12181:
12177:
12173:
12164:on 2014-03-05
12163:
12156:
12155:
12149:
12146:
12140:
12135:
12134:
12128:
12123:
12120:
12116:
12111:
12106:
12102:
12098:
12097:
12092:
12088:
12084:
12079:
12073:
12069:
12068:
12063:
12059:
12056:
12052:
12048:
12044:
12039:
12034:
12029:
12024:
12020:
12016:
12011:
12007:
12006:
11996:
11995:
11990:
11986:
11983:
11981:0-471-31716-0
11977:
11973:
11969:
11965:
11960:
11959:
11953:
11949:
11946:on 2007-06-15
11945:
11941:
11937:
11933:
11929:
11925:
11922:
11916:
11912:
11911:
11906:
11902:
11899:
11893:
11889:
11884:
11880:
11878:3-540-41129-1
11874:
11870:
11869:Integration I
11866:
11862:
11859:
11853:
11849:
11848:
11843:
11839:
11836:
11830:
11826:
11821:
11820:
11807:
11802:
11795:
11790:
11783:
11778:
11771:
11766:
11759:
11754:
11747:
11742:
11735:
11730:
11723:
11718:
11711:
11706:
11699:
11694:
11687:
11682:
11675:
11670:
11663:
11658:
11651:
11646:
11639:
11634:
11627:
11622:
11615:
11610:
11603:
11598:
11591:
11586:
11579:
11574:
11567:
11562:
11555:
11550:
11548:
11540:
11535:
11528:
11523:
11516:
11511:
11504:
11499:
11492:
11487:
11480:
11475:
11469:, p. 63.
11468:
11463:
11461:
11454:, p. 81.
11453:
11448:
11442:, p. 54.
11441:
11436:
11430:, p. 80.
11429:
11424:
11422:
11415:, p. 25.
11414:
11409:
11407:
11400:, p. 53.
11399:
11394:
11388:, p. 14.
11387:
11382:
11375:
11374:Bourbaki 2004
11370:
11363:
11358:
11351:
11346:
11339:
11334:
11327:
11322:
11315:
11310:
11304:, p. 69.
11303:
11298:
11291:
11286:
11280:, p. 74.
11279:
11274:
11272:
11264:
11259:
11252:
11247:
11240:
11236:
11231:
11224:
11220:
11215:
11208:
11203:
11196:
11191:
11184:
11179:
11172:
11167:
11160:
11155:
11148:
11143:
11136:
11131:
11124:
11119:
11111:
11107:
11103:
11099:
11095:
11091:
11087:
11080:
11073:
11068:
11061:
11056:
11049:
11044:
11037:
11032:
11028:
11014:
11010:
11004:
11000:
10990:
10987:
10984:
10981:
10978:
10975:
10974:
10970:
10964:
10959:
10938:
10935:
10922:
10916:
10913:
10910:
10900:
10894:
10888:
10885:
10882:
10879:
10874:
10871:
10868:
10863:
10860:
10857:
10842:
10836:
10833:
10830:
10827:
10824:
10821:
10814:
10808:
10805:
10800:
10795:
10791:
10783:
10782:
10781:
10779:
10764:
10762:
10758:
10740:
10727:
10723:
10719:
10709:
10707:
10703:
10700:
10694:
10684:
10682:
10678:
10668:
10666:
10661:
10657:
10647:
10643:
10637:
10632:
10626:
10622:
10614:
10610:
10600:
10596:
10586:
10582:
10577:
10573:
10571:
10567:
10563:
10558:
10549:. The degree
10548:
10543:
10541:
10537:
10533:
10529:
10520:
10515:
10505:
10503:
10498:
10492:
10490:
10486:
10482:
10478:
10474:
10470:
10466:
10464:
10458:
10456:
10452:
10448:
10444:
10440:
10435:
10433:
10429:
10425:
10421:
10416:
10412:
10408:
10404:
10400:
10396:
10392:
10388:
10383:
10381:
10377:
10373:
10369:
10363:
10353:
10351:
10346:
10344:
10340:
10336:
10331:
10329:
10325:
10321:
10317:
10316:Taylor series
10313:
10308:
10306:
10302:
10298:
10294:
10274:
10268:
10262:
10259:
10253:
10247:
10244:
10241:
10238:
10231:
10225:
10220:
10215:
10211:
10203:
10202:
10201:
10199:
10198:singularities
10194:
10190:
10171:
10167:
10162:
10147:
10145:
10141:
10122:
10119:
10116:
10109:
10103:
10098:
10093:
10089:
10085:
10080:
10074:
10070:
10062:
10061:
10060:
10046:
10026:
10003:
9997:
9974:
9968:
9945:
9942:
9939:
9932:
9926:
9921:
9916:
9912:
9908:
9902:
9896:
9893:
9887:
9881:
9874:
9873:
9872:
9855:
9852:
9849:
9838:
9834:
9830:
9826:
9822:
9806:
9803:
9800:
9793:
9785:
9781:
9775:
9770:
9766:
9762:
9753:
9747:
9741:
9734:
9730:
9724:
9707:
9687:
9682:
9678:
9674:
9666:
9661:
9657:
9652:
9650:
9646:
9642:
9638:
9628:
9626:
9622:
9616:
9606:
9604:
9600:
9596:
9590:
9580:
9578:
9574:
9570:
9566:
9562:
9558:
9554:
9548:
9529:
9526:
9518:
9515:
9507:
9498:
9497:cross product
9494:
9493:wedge product
9478:
9458:
9455:
9452:
9449:
9446:
9443:
9440:
9437:
9434:
9431:
9428:
9425:
9422:
9419:
9416:
9413:
9410:
9387:
9384:
9381:
9378:
9375:
9372:
9365:
9362:
9359:
9356:
9353:
9347:
9344:
9341:
9338:
9335:
9332:
9329:
9322:
9319:
9316:
9313:
9310:
9304:
9301:
9298:
9295:
9292:
9289:
9286:
9279:
9276:
9273:
9270:
9267:
9261:
9254:
9253:
9252:
9249:
9247:
9243:
9239:
9234:
9230:
9226:
9207:
9204:
9197:
9194:
9191:
9188:
9185:
9179:
9176:
9173:
9170:
9163:
9160:
9157:
9154:
9151:
9145:
9142:
9139:
9136:
9129:
9126:
9123:
9120:
9117:
9111:
9104:
9103:
9102:
9100:
9096:
9092:
9088:
9082:
9078:
9072:
9062:
9060:
9041:
9038:
9035:
9028:
9022:
9017:
9013:
9005:
9004:
9003:
8989:
8973:
8969:
8963:
8953:
8951:
8947:
8928:
8918:
8914:
8902:
8898:
8890:
8889:
8888:
8885:
8880:
8875:
8870:
8865:
8860:
8850:
8846:
8840:
8833:
8829:
8823:
8817:
8811:
8807:on a surface
8805:
8799:
8797:
8793:
8789:
8788:line integral
8785:
8781:
8777:
8773:
8764:
8746:
8738:
8735:
8725:
8721:
8717:
8714:
8707:
8706:
8705:
8702:
8699:
8695:
8689:
8684:
8680:
8675:
8671:
8666:
8646:
8638:
8630:
8627:
8620:
8619:
8618:
8615:
8609:
8604:
8600:
8596:
8592:
8588:
8584:
8580:
8576:
8571:
8569:
8565:
8561:
8557:
8556:path integral
8553:
8552:line integral
8548:
8546:
8545:vector fields
8536:
8531:
8527:
8526:Line integral
8503:
8500:
8497:
8493:
8488:
8484:
8480:
8467:
8463:
8448:
8443:
8439:
8431:
8430:
8429:
8413:
8398:
8394:
8390:
8371:
8368:
8365:
8358:
8355:
8352:
8346:
8341:
8337:
8329:
8328:
8327:
8325:
8306:
8303:
8300:
8295:
8291:
8288:
8281:
8278:
8275:
8269:
8264:
8259:
8255:
8250:
8244:
8239:
8235:
8227:
8226:
8225:
8223:
8219:
8215:
8209:
8205:
8201:
8197:
8192:
8188:
8183:
8163:
8160:
8153:
8150:
8147:
8141:
8136:
8132:
8124:
8123:
8122:
8105:
8102:
8099:
8093:
8087:
8084:
8081:
8075:
8072:
8064:
8061:given as the
8060:
8056:
8052:
8048:
8044:
8040:
8036:
8032:
8010:
8007:
8004:
7998:
7995:
7992:
7983:
7978:
7968:
7958:
7954:
7935:
7932:
7929:
7922:
7916:
7911:
7906:
7903:
7900:
7896:
7890:
7884:
7876:
7873:
7870:
7863:
7857:
7852:
7847:
7843:
7835:
7834:
7833:
7829:
7825:
7804:
7801:
7798:
7791:
7785:
7780:
7775:
7771:
7759:
7751:
7748:
7745:
7738:
7732:
7722:
7718:
7710:
7709:
7708:
7705:
7703:
7699:
7695:
7674:
7671:
7663:
7655:
7652:
7649:
7641:
7638:
7625:
7621:
7613:
7608:
7603:
7574:
7568:
7562:
7559:
7553:
7547:
7544:
7541:
7538:
7531:
7525:
7520:
7515:
7511:
7503:
7502:
7501:
7480:
7474:
7467:
7464:
7460:
7454:
7448:
7441:
7440:
7439:
7422:
7418:
7404:
7400:
7396:
7368:
7362:
7359:
7353:
7346:
7343:
7335:
7334:
7333:
7329:
7325:
7300:
7297:
7294:
7287:
7281:
7276:
7271:
7267:
7263:
7257:
7251:
7244:
7243:
7242:
7232:
7221:First theorem
7218:
7216:
7212:
7208:
7204:
7198:
7188:
7155:
7152:
7145:
7139:
7134:
7129:
7125:
7121:
7118:
7115:
7108:
7102:
7097:
7092:
7088:
7084:
7080:
7072:
7069:
7062:
7056:
7051:
7046:
7042:
7038:
7035:
7032:
7025:
7019:
7014:
7009:
7005:
7001:
6997:
6992:
6989:
6982:
6976:
6971:
6966:
6962:
6950:
6949:
6948:
6931:
6928:
6925:
6918:
6912:
6907:
6902:
6898:
6894:
6891:
6888:
6881:
6875:
6870:
6865:
6861:
6857:
6854:
6851:
6844:
6838:
6833:
6828:
6824:
6816:
6815:
6814:
6813:
6810:
6794:
6790:
6771:
6768:
6765:
6762:
6755:
6749:
6744:
6739:
6735:
6727:
6726:
6725:
6722:
6718:
6698:
6695:
6692:
6685:
6679:
6674:
6669:
6665:
6661:
6658:
6655:
6652:
6645:
6639:
6634:
6629:
6625:
6617:
6616:
6615:
6614:
6610:
6606:
6597:
6593:
6575:
6571:
6567:whose values
6565:
6560:
6556:
6549:
6542:
6538:
6528:
6524:
6504:
6501:
6494:
6488:
6483:
6478:
6474:
6466:
6465:
6464:
6462:
6458:
6441:
6425:
6420:
6416:
6412:
6407:
6403:
6400:
6394:
6389:
6382:
6376:
6372:
6367:
6363:
6358:
6353:
6349:
6345:
6340:
6336:
6333:
6327:
6322:
6315:
6309:
6305:
6300:
6296:
6291:
6286:
6282:
6278:
6273:
6269:
6266:
6260:
6255:
6248:
6242:
6239:
6233:
6227:
6223:
6218:
6214:
6204:
6197:
6193:
6183:
6177:
6160:
6155:
6152:
6147:
6143:
6126:
6121:
6117:
6113:
6108:
6104:
6101:
6095:
6090:
6083:
6077:
6073:
6068:
6064:
6057:
6053:
6049:
6044:
6040:
6037:
6031:
6026:
6019:
6013:
6009:
6004:
6000:
5995:
5991:
5987:
5984:
5977:
5971:
5965:
5959:
5956:
5952:
5943:
5936:
5926:
5905:
5887:
5870:
5866:
5852:
5849:
5844:
5840:
5826:
5822:
5821:inner product
5818:
5817:Hilbert space
5814:
5798:
5794:
5790:
5787:
5781:
5773:
5767:
5762:
5757:
5753:
5748:
5743:
5739:
5736:
5730:
5722:
5716:
5711:
5706:
5702:
5697:
5693:
5688:
5683:
5679:
5676:
5669:
5660:
5657:
5649:
5644:
5640:
5635:
5624:
5612:Moreover, if
5599:
5596:
5593:
5581:
5575:
5565:
5560:
5556:
5552:
5548:
5544:
5541:
5534:
5528:
5523:
5518:
5514:
5509:
5497:
5488:
5484:
5462:
5451:
5445:
5437:
5431:
5420:
5411:
5406:
5395:
5389:
5383:
5377:
5369:
5365:
5360:
5354:
5348:
5342:
5336:
5333:
5327:
5318:
5315:
5304:
5300:
5288:
5285:
5271:
5268:
5265:
5258:
5252:
5247:
5242:
5238:
5234:
5231:
5228:
5221:
5215:
5210:
5205:
5201:
5186:
5182:
5175:
5171:
5163:
5159:
5153:
5152:Subintervals.
5150:
5136:
5133:
5130:
5123:
5117:
5112:
5107:
5103:
5099:
5096:
5093:
5086:
5080:
5075:
5070:
5066:
5055:
5051:
5039:
5035:
5031:
5027:
5020:
5016:
5004:
5000:
4996:
4979:
4976:
4973:
4966:
4960:
4955:
4950:
4946:
4942:
4939:
4936:
4929:
4923:
4918:
4913:
4909:
4890:
4886:
4874:
4870:
4866:
4862:
4857:
4854:
4840:
4834:
4831:
4828:
4822:
4819:
4816:
4813:
4806:
4800:
4795:
4790:
4786:
4782:
4776:
4773:
4770:
4764:
4754:
4750:
4746:
4739:
4735:
4731:
4724:
4720:
4708:
4704:
4693:
4689:
4685:
4681:
4669:
4665:
4659:
4655:
4645:
4642:
4641:
4640:
4636:
4632:
4627:
4624:
4620:
4617:defined on a
4616:
4606:
4604:
4600:
4592:
4586:
4584:
4583:Hilbert space
4581:is a complex
4575:
4571:
4558:
4552:
4548:
4542:
4536:
4507:
4504:
4501:
4497:
4492:
4488:
4481:
4474:
4473:
4472:
4469:
4455:
4451:
4447:
4443:
4439:
4432:
4429:
4426:
4420:
4416:
4410:
4391:
4388:
4385:
4381:
4376:
4372:
4368:
4365:
4362:
4359:
4355:
4350:
4346:
4342:
4339:
4336:
4333:
4326:
4323:
4320:
4317:
4314:
4306:
4302:
4294:
4293:
4292:
4275:
4272:
4268:
4263:
4259:
4252:
4245:
4244:
4243:
4238:with measure
4234:
4233:measure space
4230:
4210:
4207:
4204:
4197:
4191:
4186:
4181:
4177:
4173:
4170:
4167:
4164:
4157:
4151:
4146:
4141:
4137:
4133:
4130:
4127:
4124:
4117:
4108:
4105:
4102:
4099:
4096:
4088:
4083:
4079:
4071:
4070:
4069:
4067:
4063:
4044:
4041:
4034:
4028:
4023:
4018:
4014:
4007:
4000:
3999:
3998:
3996:
3992:
3974:
3970:
3966:
3963:
3959:
3956:
3952:
3948:
3944:
3941:
3937:
3933:
3930:
3926:
3922:
3918:
3914:
3911:
3907:
3903:
3899:
3898:Arnaud Denjoy
3895:
3891:
3888:
3884:
3883:Haar integral
3880:
3877:
3873:
3869:
3865:
3862:
3858:
3854:
3851:
3847:
3844:
3840:
3836:
3835:
3834:
3794:
3791:
3784:
3781:
3778:
3772:
3766:
3756:
3750:
3744:
3741:
3735:
3728:
3724:
3715:
3712:
3706:
3700:
3697:
3686:
3682:
3673:
3665:
3661:
3638:
3635:
3628:
3625:
3622:
3616:
3610:
3600:
3594:
3588:
3582:
3575:
3571:
3562:
3559:
3553:
3547:
3536:
3532:
3523:
3515:
3511:
3498:
3497:
3496:
3479:
3476:
3470:
3466:
3460:
3456:
3452:
3449:
3446:
3440:
3436:
3430:
3426:
3422:
3419:
3416:
3412:
3407:
3403:
3395:
3394:
3393:
3368:
3362:
3359:
3356:
3353:
3344:
3334:
3330:
3322:
3321:
3320:
3305:
3303:
3299:
3294:
3274:
3271:
3264:
3256:
3252:
3241:
3237:
3233:
3230:
3227:
3220:
3219:
3218:
3210:
3206:
3202:
3198:
3194:
3190:
3186:
3180:
3176:
3172:
3168:
3164:
3160:
3154:
3150:
3146:
3140:
3136:
3126:
3122:
3118:
3108:
3105:
3101:
3094:
3087:
3083:
3079:
3075:
3065:
3061:
3049:
3044:
3042:
3038:
3032:
3024:
3019:
3009:
3007:
3003:
2984:
2981:
2978:
2974:
2968:
2954:
2950:
2943:
2938:
2933:
2930:
2927:
2923:
2919:
2916:
2912:
2904:
2903:
2887:
2864:
2861:
2858:
2835:
2832:
2829:
2822:there exists
2809:
2806:
2803:
2795:
2794:
2793:
2783:
2777:
2771:
2767:
2760:
2753:
2749:
2744:
2740:
2735:
2713:
2708:
2694:
2690:
2683:
2678:
2673:
2670:
2667:
2663:
2655:
2654:
2653:
2647:
2640:
2636:
2606:
2603:
2600:
2595:
2591:
2587:
2582:
2578:
2574:
2569:
2566:
2563:
2559:
2555:
2552:
2549:
2544:
2540:
2536:
2531:
2527:
2523:
2518:
2514:
2510:
2505:
2501:
2497:
2492:
2488:
2484:
2481:
2474:
2473:
2472:
2470:
2466:
2462:
2456:
2446:
2433:
2413:
2409:
2401:
2392:
2388:
2380:
2366:
2363:
2355:
2328:
2323:
2320:
2315:
2312:
2309:
2303:
2296:
2291:
2287:
2279:
2278:
2277:
2254:
2251:
2248:
2244:
2238:
2235:
2230:
2225:
2222:
2216:
2209:
2206:
2200:
2197:
2194:
2190:
2184:
2181:
2176:
2171:
2168:
2162:
2155:
2152:
2146:
2142:
2138:
2135:
2130:
2127:
2121:
2114:
2111:
2100:
2099:
2098:
2060:
2053:
2034:
2023:
2019:
2013:
2011:
2010:infinitesimal
1999:
1990:
1981:
1967:
1962:
1957:
1953:
1947:
1943:
1939:
1936:
1931:
1926:
1922:
1916:
1912:
1908:
1902:
1897:
1893:
1889:
1886:
1881:
1877:
1868:
1863:
1859:
1849:
1843:
1840:
1836:
1817:
1814:
1811:
1804:
1798:
1795:
1788:
1787:
1786:
1783:
1779:
1776:
1770:
1763:
1759:
1753:
1747:
1742:
1739:, called the
1737:
1713:
1710:
1707:
1700:
1694:
1689:
1684:
1680:
1672:
1671:
1670:
1667:
1660:
1656:
1652:
1642:
1640:
1630:
1626:
1617:
1612:
1604:
1600:
1598:
1594:
1590:
1586:
1582:
1578:
1574:
1570:
1566:
1556:
1554:
1550:
1549:standard part
1546:
1545:real analysis
1542:
1539:, founded in
1538:
1535:formulated a
1534:
1530:
1526:
1522:
1518:
1514:
1510:
1506:
1499:Formalization
1496:
1494:
1490:
1486:
1482:
1472:
1466:
1462:
1458:
1454:
1450:
1445:
1443:
1439:
1435:
1431:
1427:
1423:
1417:
1413:up to degree
1411:
1406:
1402:
1398:
1393:
1391:
1375:
1355:
1352:
1346:
1342:
1338:
1316:
1312:
1308:
1305:
1297:
1296:fourth powers
1282:
1277:
1275:
1271:
1267:
1263:
1258:
1256:
1252:
1248:
1244:
1240:
1237:, area of an
1236:
1232:
1228:
1224:
1220:
1216:
1212:
1208:
1204:
1203:ancient Greek
1200:
1189:
1179:
1177:
1173:
1169:
1165:
1161:
1160:line integral
1157:
1152:
1150:
1146:
1142:
1138:
1134:
1133:infinitesimal
1130:
1126:
1122:
1117:
1115:
1111:
1107:
1103:
1102:
1097:
1093:
1089:
1086:computes the
1085:
1080:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1038:
1033:
1031:
1026:
1024:
1019:
1018:
1016:
1015:
1008:
1005:
1003:
1000:
998:
995:
993:
990:
988:
985:
983:
980:
978:
975:
974:
966:
965:
958:
955:
953:
950:
948:
945:
943:
940:
939:
931:
930:
919:
916:
914:
911:
909:
906:
905:
904:
903:
893:
892:
881:
878:
876:
873:
871:
868:
866:
863:
861:
860:Line integral
858:
856:
853:
851:
848:
847:
846:
845:
841:
840:
835:
832:
830:
827:
825:
822:
820:
817:
816:
815:
814:
810:
809:
803:
802:Multivariable
797:
796:
785:
782:
780:
777:
775:
772:
770:
767:
765:
762:
760:
757:
756:
755:
754:
750:
749:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
715:
714:
713:
707:
701:
700:
689:
686:
684:
681:
679:
676:
674:
671:
669:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
639:
638:
637:
634:
631:
630:
625:
622:
620:
617:
615:
612:
610:
607:
605:
602:
599:
595:
592:
591:
590:
589:
583:
577:
576:
565:
562:
560:
557:
555:
552:
550:
547:
544:
540:
537:
535:
532:
529:
525:
521:
520:trigonometric
517:
514:
512:
509:
507:
504:
502:
499:
498:
497:
496:
492:
491:
486:
483:
481:
478:
476:
473:
471:
468:
465:
461:
458:
456:
453:
452:
451:
450:
446:
445:
440:
437:
435:
432:
430:
427:
426:
425:
424:
418:
412:
411:
400:
397:
395:
392:
390:
387:
385:
382:
380:
377:
375:
372:
370:
367:
365:
362:
360:
357:
355:
352:
351:
350:
349:
346:
343:
342:
337:
334:
332:
331:Related rates
329:
327:
324:
322:
319:
317:
314:
312:
309:
308:
307:
306:
302:
301:
294:
291:
289:
288:of a function
286:
284:
283:infinitesimal
281:
280:
279:
276:
273:
269:
266:
265:
264:
263:
259:
258:
252:
246:
245:
239:
236:
234:
231:
229:
226:
225:
220:
217:
215:
212:
211:
207:
204:
203:
202:
201:
182:
176:
173:
167:
161:
158:
155:
152:
145:
138:
135:
129:
124:
120:
111:
110:
107:
104:
103:
99:
98:
77:
71:
63:
58:
52:
48:
41:
37:
33:
19:
13770:
13597:
13591:
13338:Inequalities
13278:Uniform norm
13166:
13135:
13086:
12955:Itô integral
12790:Substitution
12781:Integration
12680:
12642:P. S. Wang,
12637:
12619:
12610:, link from
12586:
12577:
12556:Online books
12531:
12511:
12490:
12468:
12461:Stoer, Josef
12446:, Springer,
12441:
12417:
12404:
12382:
12348:(32): 1073,
12345:
12341:
12322:
12295:
12278:
12255:
12205:
12201:
12179:
12166:, retrieved
12162:the original
12153:
12132:
12127:Moler, Cleve
12100:
12094:
12066:
12062:Heath, T. L.
12018:
12014:
12004:
11993:
11971:
11957:
11944:the original
11935:
11909:
11887:
11868:
11846:
11824:
11816:Bibliography
11801:
11789:
11777:
11765:
11753:
11741:
11729:
11717:
11710:Apostol 1967
11705:
11693:
11686:Apostol 1967
11681:
11674:Apostol 1967
11669:
11664:, p. 3.
11657:
11652:, p. 1.
11645:
11633:
11621:
11609:
11597:
11585:
11573:
11561:
11534:
11527:Apostol 1967
11522:
11515:Apostol 1967
11510:
11498:
11491:Apostol 1967
11486:
11479:Apostol 1967
11474:
11452:Apostol 1967
11447:
11435:
11428:Apostol 1967
11398:Folland 1999
11393:
11381:
11369:
11362:Folland 1999
11357:
11345:
11340:, p. 5.
11333:
11321:
11309:
11302:Apostol 1967
11297:
11285:
11278:Apostol 1967
11258:
11246:
11239:Fourier 1822
11230:
11223:Leibniz 1899
11214:
11202:
11190:
11178:
11166:
11154:
11142:
11130:
11118:
11093:
11089:
11079:
11067:
11055:
11043:
11031:
11009:Apostol 1967
11003:
10775:
10715:
10696:
10674:
10662:
10655:
10635:
10624:
10620:
10612:
10608:
10598:
10594:
10584:
10580:
10574:
10554:
10544:
10525:
10493:
10488:
10484:
10480:
10476:
10472:
10462:
10459:
10436:
10384:
10365:
10347:
10332:
10309:
10289:
10192:
10188:
10169:
10165:
10158:
10137:
9960:
9871:is given by
9825:displacement
9751:
9745:
9739:
9732:
9728:
9722:
9653:
9634:
9631:Applications
9618:
9592:
9549:
9491:denotes the
9402:
9250:
9245:
9241:
9237:
9232:
9228:
9224:
9222:
9084:
9056:
8965:
8943:
8883:
8873:
8863:
8848:
8844:
8842:, such that
8838:
8831:
8827:
8821:
8815:
8809:
8803:
8800:
8796:vector field
8792:scalar field
8771:
8769:
8700:
8697:
8693:
8687:
8673:
8670:vector field
8664:
8661:
8613:
8607:
8601:is equal to
8591:differential
8579:vector field
8575:scalar field
8572:
8567:
8555:
8551:
8549:
8541:
8396:
8392:
8388:
8386:
8323:
8321:
8217:
8213:
8207:
8203:
8199:
8195:
8191:Riemann sums
8181:
8178:
8058:
8054:
8050:
8046:
8038:
8034:
8028:
7950:
7827:
7823:
7819:
7706:
7690:
7495:
7410:
7398:
7394:
7386:
7327:
7323:
7315:
7224:
7214:
7202:
7200:
7174:
6946:
6812:
6791:, should be
6786:
6720:
6716:
6713:
6613:
6608:
6604:
6573:
6569:
6563:
6558:
6554:
6547:
6540:
6536:
6526:
6522:
6519:
6450:
6195:
6191:
6181:
6175:
6158:
6153:
6145:
6141:
5934:
5924:
5903:
5885:
5868:
5864:
5850:
5842:
5838:
5622:
5495:
5486:
5482:
5286:
5184:
5180:
5173:
5169:
5161:
5157:
5151:
5053:
5049:
5037:
5033:
5029:
5025:
5018:
5014:
5002:
4998:
4994:
4888:
4884:
4872:
4868:
4864:
4860:
4855:
4752:
4748:
4744:
4737:
4733:
4729:
4722:
4718:
4706:
4702:
4691:
4687:
4683:
4679:
4668:real numbers
4657:
4653:
4643:
4634:
4630:
4612:
4609:Inequalities
4587:
4573:
4569:
4550:
4546:
4540:
4534:
4523:
4467:
4453:
4449:
4445:
4441:
4418:
4414:
4406:
4290:
4226:
4059:
3991:vector space
3988:
3973:Banach space
3917:Itô integral
3902:Oskar Perron
3861:Johann Radon
3832:
3494:
3383:
3306:
3298:well-defined
3292:
3289:
3208:
3204:
3200:
3196:
3192:
3188:
3184:
3178:
3174:
3170:
3166:
3162:
3158:
3152:
3148:
3144:
3138:
3134:
3124:
3120:
3116:
3109:
3103:
3099:
3092:
3085:
3081:
3063:
3059:
3051:
3046:
3033:
3029:
2999:
2781:
2775:
2769:
2765:
2758:
2751:
2747:
2742:
2738:
2733:
2728:
2645:
2638:
2634:
2623:
2464:
2461:Riemann sums
2458:
2443:
2411:
2407:
2390:
2386:
2371:Darboux sums
2361:
2353:
2344:which means
2343:
2270:
2058:
2051:
2021:
2017:
2014:
2006:
1997:
1847:
1844:
1832:
1784:
1774:
1768:
1761:
1757:
1751:
1745:
1741:differential
1735:
1728:
1665:
1658:
1654:
1648:
1636:
1624:
1615:
1610:
1603:Isaac Newton
1601:
1596:
1588:
1587:(written as
1584:
1576:
1572:
1562:
1502:
1478:
1446:
1425:
1415:
1409:
1394:
1278:
1259:
1227:surface area
1214:
1196:
1153:
1125:Isaac Newton
1118:
1116:operations.
1105:
1099:
1083:
1081:
1052:
1046:
516:Substitution
459:
416:
278:Differential
251:Differential
13598:Integration
13536:Von Neumann
13350:Chebyshev's
12805:Weierstrass
11662:Feller 1966
11650:Feller 1966
11326:Krantz 1991
11263:Cajori 1929
11251:Cajori 1929
11235:Cajori 1929
11219:Burton 2011
11171:Burton 2011
11147:Struik 1986
11036:Burton 2011
10699:geometrical
10687:Geometrical
10631:interpolate
10618:is half of
10420:Mathematica
10401:functions,
10399:exponential
10387:closed form
10150:Computation
9077:Volume form
8869:dot product
8677:such as an
8037:-axis, the
7957:real number
6457:real-valued
6447:Conventions
4567:, and when
3887:Alfréd Haar
3041:Paul Montel
3002:Darboux sum
2646:Riemann sum
1424:. The case
1292: 1040
1270:Zu Chongzhi
1251:hyperboloid
1205:astronomer
1141:curvilinear
1088:signed area
1049:mathematics
977:Precalculus
970:Miscellanea
935:Specialized
842:Definitions
609:Alternating
447:Definitions
260:Definitions
62:signed area
14188:Categories
14143:Similarity
14085:Perplexity
13996:Rand index
13981:Dunn index
13966:Silhouette
13958:Clustering
13822:Regression
13623:stochastic
13531:C*-algebra
13355:Clarkson's
12920:incomplete
12783:techniques
12612:HathiTrust
12528:"Integral"
12219:1507.04348
12212:: 012015,
12168:2014-02-28
12028:1707.08942
11467:Rudin 1987
11440:Rudin 1987
11413:Rudin 1987
11338:Rudin 1987
11048:Heath 2002
11023:References
10677:planimeter
10671:Mechanical
10441:(like the
10426:and other
10393:, include
10186:such that
10155:Analytical
9821:kinematics
9583:Summations
9575:, and the
9075:See also:
8583:arc length
7700:of proper
7591:Extensions
6811:of , then:
6553:≤ . . . ≤
5827:functions
3980:Properties
3947:rough path
3801:otherwise.
3645:otherwise,
1780:integrable
1453:Torricelli
1438:quadrature
1390:paraboloid
1290: – c.
1288: 965
1247:paraboloid
1219:Archimedes
1211:Democritus
1186:See also:
957:Variations
952:Stochastic
942:Fractional
811:Formalisms
774:Divergence
743:Identities
723:Divergence
268:Derivative
219:Continuity
14194:Integrals
13913:Precision
13865:RMSE/RMSD
13735:Functions
13526:*-algebra
13511:Quasinorm
13380:Minkowski
13271:Essential
13234:∞
13063:Lp spaces
12690:integrals
12688:Types of
12681:Integrals
12538:EMS Press
12244:119642596
12119:0273-0979
12070:, Dover,
12055:222004668
12047:2391-5455
11207:Katz 2009
11195:Katz 2009
11159:Katz 2009
11135:Katz 2009
11123:Katz 2009
11110:1573-1340
11072:Katz 2009
11060:Katz 2009
10917:
10911:−
10901:−
10895:π
10889:
10883:−
10875:π
10837:
10831:−
10809:
10801:π
10792:∫
10737:∂
10728:operator
10681:displaced
10508:Numerical
10403:logarithm
10260:−
10212:∫
10090:∫
10078:→
9913:∫
9894:−
9767:∫
9763:π
9675:π
9595:summation
9479:∧
9453:∧
9435:∧
9417:∧
9379:∧
9336:∧
9293:∧
9018:γ
9014:∫
8990:γ
8915:⋅
8899:∫
8736:⋅
8722:∫
8639:⋅
8595:intervals
8485:∫
8440:∫
8338:∬
8256:∫
8236:∫
8133:∫
8094:×
7907:ϵ
7897:∫
7888:→
7885:ε
7844:∫
7772:∫
7766:∞
7763:→
7728:∞
7719:∫
7675:π
7631:∞
7622:∫
7560:−
7512:∫
7268:∫
7126:∫
7089:∫
7043:∫
7039:−
7006:∫
6963:∫
6899:∫
6862:∫
6825:∫
6736:∫
6666:∫
6662:−
6626:∫
6475:∫
6368:∫
6301:∫
6292:≤
6219:∫
6069:∫
6005:∫
5996:≤
5957:∫
5754:∫
5703:∫
5694:≤
5641:∫
5557:∫
5553:≤
5515:∫
5239:∫
5235:≤
5202:∫
5104:∫
5067:∫
5042:for each
4947:∫
4943:≤
4910:∫
4877:for each
4832:−
4820:≤
4787:∫
4783:≤
4774:−
4615:functions
4505:μ
4489:∫
4485:↦
4389:μ
4373:∫
4369:β
4363:μ
4347:∫
4343:α
4337:μ
4324:β
4315:α
4303:∫
4276:μ
4260:∫
4256:↦
4178:∫
4174:β
4138:∫
4134:α
4106:β
4097:α
4080:∫
4015:∫
4011:↦
3985:Linearity
3742:−
3698:−
3666:−
3480:μ
3471:−
3457:∫
3453:−
3450:μ
3427:∫
3420:μ
3404:∫
3366:∞
3357:μ
3331:∫
3257:∗
3247:∞
3238:∫
3228:∫
2982:ε
2965:Δ
2924:∑
2920:−
2888:δ
2830:δ
2804:ε
2705:Δ
2664:∑
2588:≤
2575:≤
2567:−
2556:≤
2553:⋯
2550:≤
2537:≤
2524:≤
2511:≤
2498:≤
2288:∫
2249:≈
2231:−
2198:⋯
2177:−
2136:−
1954:∫
1923:∫
1860:∫
1796:∫
1681:∫
1525:piecewise
1444:in 1647.
1442:hyperbola
1399:with his
1397:Cavalieri
1339:∫
1096:real line
947:Malliavin
834:Geometric
733:Laplacian
683:Dirichlet
594:Geometric
174:−
121:∫
14129:Coverage
13908:Accuracy
13760:Infinity
13613:ordinary
13593:Calculus
13375:Markov's
13370:Hölder's
13360:Hanner's
13177:Bessel's
13115:function
13099:Lebesgue
13010:Volumes
12915:complete
12812:By parts
12587:Calculus
12438:(1989),
12402:(1964),
12372:56487062
12296:Analysis
12294:(2001),
12253:(1991),
12178:(2009),
12089:(1953),
11991:(1822),
11970:(1999),
11954:(1966),
11907:(1929),
11867:(2004),
11844:(1967),
10955:See also
10767:Examples
10724:and the
10432:radicals
10395:rational
10356:Symbolic
10142:, where
9833:velocity
9700:, where
9649:function
9557:gradient
8560:function
7698:sequence
7468:′
7387:for all
7347:′
7241:in , by
6592:interval
6461:function
6440:L spaces
5183: (
4695:for all
4677:so that
4626:interval
4448: :
4428:complete
3927:such as
3889:in 1933.
3876:measures
3763:if
3607:if
3315:and the
3199: :
3165: :
3119: :
2796:For all
2049:between
1597:Mémoires
1555:system.
1533:Lebesgue
1493:calculus
1430:function
1243:parabola
1164:interval
1069:calculus
1053:integral
987:Glossary
897:Advanced
875:Jacobian
829:Exterior
759:Gradient
751:Theorems
718:Gradient
657:Integral
619:Binomial
604:Harmonic
464:improper
460:Integral
417:Integral
399:Reynolds
374:Quotient
303:Concepts
139:′
106:Calculus
14021:Ranking
14011:SimHash
13898:F-score
13618:partial
13395:Results
13094:Measure
13014:Washers
12540:, 2001
12350:Bibcode
12224:Bibcode
11241:, §231.
10376:Macsyma
9099:tensors
8776:surface
6809:element
6807:is any
6205:holds:
5944:holds:
5908:
5894:
5890:
5876:
5823:of two
5193:, then
5058:, then
5032:) <
4901:. Thus
4664:bounded
4623:bounded
4591:Daniell
3392:-axis:
3207:) >
3173:) >
3074:measure
2351:√
2091:√
2089:, ...,
2084:√
2077:√
2070:√
1995:√
1521:Riemann
1485:Leibniz
1440:of the
1281:Alhazen
1274:Zu Geng
1266:Liu Hui
1239:ellipse
1207:Eudoxus
1201:of the
1182:History
1172:surface
1135:width.
1114:inverse
1077:physics
1065:volumes
982:History
880:Hessian
769:Stokes'
764:Green's
596: (
518: (
462: (
384:Inverse
359:Product
270: (
36:integer
13918:Recall
13755:Series
13248:spaces
13169:spaces
13138:spaces
13089:spaces
13077:Banach
13019:Shells
12514:, 2006
12497:
12475:
12450:
12424:
12389:
12370:
12330:
12310:
12263:
12242:
12190:
12141:
12117:
12074:
12053:
12045:
11978:
11917:
11894:
11875:
11854:
11831:
11108:
10706:square
10530:. The
10453:, the
10449:, the
10445:, the
10303:, and
10163:. Let
9961:where
9831:, and
9223:where
9097:, and
8857:. The
8478:
8043:volume
7332:, and
7316:Then,
7233:. Let
7183:, and
6199:|
6189:|
6185:|
6173:|
5938:|
5932:|
5928:|
5922:|
5912:, and
5501:, and
5499:|
5493:|
4619:closed
4559:, and
3495:where
3182:. Let
2780:. The
2757:. The
2252:0.7497
1581:long s
1517:limits
1505:rigour
1489:Newton
1465:Wallis
1449:Barrow
1432:, the
1405:Fermat
1255:spiral
1235:sphere
1231:volume
1225:, the
1156:domain
1108:. The
824:Tensor
819:Matrix
706:Vector
624:Taylor
582:Series
214:Limits
13923:Kappa
13840:sMAPE
13750:Limit
12800:Euler
12368:S2CID
12240:S2CID
12214:arXiv
12208:(1),
12158:(PDF)
12051:S2CID
12023:arXiv
10995:Notes
10648:. An
10424:Maple
10380:Maple
8794:or a
8780:space
8668:in a
8603:force
8577:or a
8564:curve
7963:, or
7959:, or
7953:limit
7696:of a
7694:limit
7438:in ,
6789:point
6714:With
6607:>
6525:<
6455:is a
5873:with
5845:]
5837:[
5489:]
5481:[
5176:]
5168:[
5164:]
5156:[
5056:]
5048:[
5021:]
5013:[
5011:over
4891:]
4883:[
4725:]
4717:[
4715:over
4709:]
4701:[
4660:]
4652:[
4637:]
4629:[
4060:is a
3066:]
3058:[
2768:=1...
1589:ſumma
1585:summa
1262:China
1233:of a
1092:graph
1061:areas
1051:, an
647:Ratio
614:Power
528:Euler
506:Discs
501:Parts
369:Power
364:Chain
293:total
14090:BLEU
14062:SSIM
14057:PSNR
14034:NDCG
13855:MSPE
13850:MASE
13845:MAPE
13433:For
13287:Maps
12495:ISBN
12473:ISBN
12448:ISBN
12422:ISBN
12387:ISBN
12328:ISBN
12308:ISBN
12261:ISBN
12188:ISBN
12139:ISBN
12115:ISSN
12072:ISBN
12043:ISSN
11976:ISBN
11940:SIAM
11915:ISBN
11892:ISBN
11873:ISBN
11852:ISBN
11829:ISBN
11106:ISSN
11011:and
10776:The
10500:the
10409:and
10397:and
10378:and
10191:′ =
10059:is:
9829:time
9749:and
9656:area
9561:curl
9559:and
9079:and
8859:flux
8599:work
8528:and
8049:and
8031:area
7610:The
7430:and
7411:Let
7225:Let
7201:The
6793:zero
6586:and
6187:and
6167:and
6139:For
5930:and
5916:and
5863:1 ≤
5857:and
5831:and
5616:and
5293:and
5178:and
5100:<
4867:) ≤
4742:and
4690:) ≤
4673:and
4621:and
4577:and
4229:real
3967:The
3960:The
3945:The
3934:The
3919:and
3915:The
3892:The
3881:The
3866:The
3855:The
3848:The
3837:The
3779:<
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3360:<
3191:) =
3142:and
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2833:>
2807:>
2792:if:
2759:mesh
2644:. A
2056:and
1772:and
1487:and
1451:and
1272:and
1229:and
1127:and
728:Curl
688:Abel
652:Root
14111:FID
14077:NLP
14067:IoU
14029:MRR
14006:SMC
13938:ROC
13933:AUC
13928:MCC
13880:MAD
13875:MDA
13860:RMS
13835:MAE
13830:MSE
12636:at
12512:W3C
12358:doi
12232:doi
12206:626
12105:doi
12033:doi
11098:doi
10914:cos
10886:cos
10834:cos
10806:sin
10658:− 1
10638:(0)
10341:or
9601:or
8966:In
8948:of
8881:to
8871:of
8819:in
8691:to
8681:or
8399:of
7881:lim
7756:lim
7496:If
7391:in
6598:of
6161:≥ 1
6148:= 2
5910:= 1
5871:≤ ∞
5475:If
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4881:in
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3542:max
2763:max
2346:2/3
2274:2/3
2086:2/5
2079:1/5
2061:= 1
2054:= 0
2024:) =
1622:or
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1420:in
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1174:in
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354:Sum
14190::
14039:AP
13903:P4
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12536:,
12530:,
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11546:^
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11420:^
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11270:^
11104:.
11092:.
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8825:,
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6772:0.
6719:=
6562:=
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5867:,
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5623:fg
5485:,
5305::
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4751:−
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4682:≤
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4572:=
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3147:=
3137:=
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2754:−1
2737:=
2642:∈
2410:=
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1507:.
1463:.
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1257:.
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9859:]
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9850:a
9847:[
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9801:d
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9786:2
9782:f
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9746:a
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9733:x
9731:(
9729:f
9723:x
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9688:h
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9527:+
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9311:x
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9299:y
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9280:z
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9225:E
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9177:+
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9112:E
9042:.
9039:z
9036:d
9032:)
9029:z
9026:(
9023:f
8980:x
8976:z
8929:.
8924:S
8919:d
8910:v
8903:S
8884:S
8874:v
8864:S
8855:x
8851:)
8849:x
8847:(
8845:v
8839:S
8834:)
8832:x
8830:(
8828:v
8822:S
8816:x
8810:S
8804:v
8747:.
8743:s
8739:d
8732:F
8726:C
8718:=
8715:W
8701:s
8698:d
8694:s
8688:s
8674:F
8665:C
8647:.
8643:s
8635:F
8631:=
8628:W
8614:s
8608:F
8504:.
8501:V
8498:d
8494:f
8489:D
8481:=
8474:x
8468:n
8464:d
8460:)
8456:x
8452:(
8449:f
8444:D
8414:n
8409:R
8397:D
8389:f
8372:.
8369:A
8366:d
8362:)
8359:y
8356:,
8353:x
8350:(
8347:f
8342:R
8324:R
8307:.
8304:x
8301:d
8296:]
8292:y
8289:d
8285:)
8282:y
8279:,
8276:x
8273:(
8270:f
8265:d
8260:c
8251:[
8245:b
8240:a
8218:f
8214:R
8210:)
8208:y
8206:,
8204:x
8202:(
8200:f
8196:z
8164:A
8161:d
8157:)
8154:y
8151:,
8148:x
8145:(
8142:f
8137:R
8109:]
8106:d
8103:,
8100:c
8097:[
8091:]
8088:b
8085:,
8082:a
8079:[
8076:=
8073:R
8059:R
8055:f
8051:y
8047:x
8035:x
8014:)
8011:y
8008:,
8005:x
8002:(
7999:f
7996:=
7993:z
7961:∞
7936:.
7933:x
7930:d
7926:)
7923:x
7920:(
7917:f
7912:b
7904:+
7901:a
7891:0
7877:=
7874:x
7871:d
7867:)
7864:x
7861:(
7858:f
7853:b
7848:a
7830:]
7828:b
7824:a
7822:(
7805:.
7802:x
7799:d
7795:)
7792:x
7789:(
7786:f
7781:b
7776:a
7760:b
7752:=
7749:x
7746:d
7742:)
7739:x
7736:(
7733:f
7723:a
7672:=
7664:x
7659:)
7656:1
7653:+
7650:x
7647:(
7642:x
7639:d
7626:0
7575:.
7572:)
7569:a
7566:(
7563:F
7557:)
7554:b
7551:(
7548:F
7545:=
7542:x
7539:d
7535:)
7532:x
7529:(
7526:f
7521:b
7516:a
7498:f
7481:.
7478:)
7475:x
7472:(
7465:F
7461:=
7458:)
7455:x
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7449:f
7436:x
7432:F
7428:f
7424:F
7413:f
7401:)
7399:b
7395:a
7393:(
7389:x
7372:)
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7360:=
7357:)
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7351:(
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7330:)
7328:b
7324:a
7322:(
7318:F
7301:.
7298:t
7295:d
7291:)
7288:t
7285:(
7282:f
7277:x
7272:a
7264:=
7261:)
7258:x
7255:(
7252:F
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7235:F
7227:f
7185:c
7181:b
7177:a
7156:x
7153:d
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7146:x
7143:(
7140:f
7135:c
7130:b
7122:+
7119:x
7116:d
7112:)
7109:x
7106:(
7103:f
7098:b
7093:a
7085:=
7073:x
7070:d
7066:)
7063:x
7060:(
7057:f
7052:b
7047:c
7036:x
7033:d
7029:)
7026:x
7023:(
7020:f
7015:b
7010:a
7002:=
6993:x
6990:d
6986:)
6983:x
6980:(
6977:f
6972:c
6967:a
6932:.
6929:x
6926:d
6922:)
6919:x
6916:(
6913:f
6908:b
6903:c
6895:+
6892:x
6889:d
6885:)
6882:x
6879:(
6876:f
6871:c
6866:a
6858:=
6855:x
6852:d
6848:)
6845:x
6842:(
6839:f
6834:b
6829:a
6805:c
6801:f
6797:f
6769:=
6766:x
6763:d
6759:)
6756:x
6753:(
6750:f
6745:a
6740:a
6721:b
6717:a
6699:.
6696:x
6693:d
6689:)
6686:x
6683:(
6680:f
6675:a
6670:b
6659:=
6656:x
6653:d
6649:)
6646:x
6643:(
6640:f
6635:b
6630:a
6612::
6609:b
6605:a
6600:f
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6580:f
6574:i
6570:x
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6559:n
6555:x
6551:1
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6544:0
6541:x
6537:a
6532:f
6527:b
6523:a
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6495:x
6492:(
6489:f
6484:b
6479:a
6453:f
6442:.
6426:.
6421:p
6417:/
6413:1
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6401:d
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6390:|
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6383:x
6380:(
6377:g
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6364:(
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6350:/
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6310:f
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6215:(
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5799:.
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5717:f
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5670:x
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5664:)
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5566:b
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5435:)
5432:x
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5412:,
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5399:)
5396:x
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5384:=
5381:)
5378:x
5375:(
5370:2
5366:f
5361:,
5358:)
5355:x
5352:(
5349:g
5346:)
5343:x
5340:(
5337:f
5334:=
5331:)
5328:x
5325:(
5322:)
5319:g
5316:f
5313:(
5295:g
5291:f
5272:.
5269:x
5266:d
5262:)
5259:x
5256:(
5253:f
5248:b
5243:a
5232:x
5229:d
5225:)
5222:x
5219:(
5216:f
5211:d
5206:c
5191:x
5187:)
5185:x
5181:f
5174:b
5170:a
5162:d
5158:c
5137:.
5134:x
5131:d
5127:)
5124:x
5121:(
5118:g
5113:b
5108:a
5097:x
5094:d
5090:)
5087:x
5084:(
5081:f
5076:b
5071:a
5054:b
5050:a
5044:x
5040:)
5038:x
5036:(
5034:g
5030:x
5028:(
5026:f
5019:b
5015:a
5009:M
5005:)
5003:a
4999:b
4997:(
4995:M
4980:.
4977:x
4974:d
4970:)
4967:x
4964:(
4961:g
4956:b
4951:a
4940:x
4937:d
4933:)
4930:x
4927:(
4924:f
4919:b
4914:a
4899:g
4895:f
4889:b
4885:a
4879:x
4875:)
4873:x
4871:(
4869:g
4865:x
4863:(
4861:f
4841:.
4838:)
4835:a
4829:b
4826:(
4823:M
4817:x
4814:d
4810:)
4807:x
4804:(
4801:f
4796:b
4791:a
4780:)
4777:a
4771:b
4768:(
4765:m
4755:)
4753:a
4749:b
4747:(
4745:M
4740:)
4738:a
4734:b
4732:(
4730:m
4723:b
4719:a
4713:f
4707:b
4703:a
4697:x
4692:M
4688:x
4686:(
4684:f
4680:m
4675:M
4671:m
4658:b
4654:a
4648:f
4635:b
4631:a
4595:X
4579:V
4574:C
4570:K
4565:K
4561:V
4551:p
4547:Q
4541:C
4535:R
4530:K
4526:V
4508:,
4502:d
4498:f
4493:E
4482:f
4468:∞
4463:V
4459:f
4454:V
4450:E
4446:f
4442:K
4434:V
4421:)
4419:μ
4417:,
4415:E
4413:(
4392:.
4386:d
4382:g
4377:E
4366:+
4360:d
4356:f
4351:E
4340:=
4334:d
4330:)
4327:g
4321:+
4318:f
4312:(
4307:E
4273:d
4269:f
4264:E
4253:f
4240:μ
4236:E
4211:.
4208:x
4205:d
4201:)
4198:x
4195:(
4192:g
4187:b
4182:a
4171:+
4168:x
4165:d
4161:)
4158:x
4155:(
4152:f
4147:b
4142:a
4131:=
4128:x
4125:d
4121:)
4118:x
4115:(
4112:)
4109:g
4103:+
4100:f
4094:(
4089:b
4084:a
4045:x
4042:d
4038:)
4035:x
4032:(
4029:f
4024:b
4019:a
4008:f
3975:.
3957:.
3942:.
3931:.
3912:.
3878:.
3795:,
3792:0
3785:,
3782:0
3776:)
3773:x
3770:(
3767:f
3757:,
3754:)
3751:x
3748:(
3745:f
3736:{
3729:=
3719:}
3716:0
3713:,
3710:)
3707:x
3704:(
3701:f
3695:{
3687:=
3677:)
3674:x
3671:(
3662:f
3639:,
3636:0
3629:,
3626:0
3620:)
3617:x
3614:(
3611:f
3601:,
3598:)
3595:x
3592:(
3589:f
3583:{
3576:=
3566:}
3563:0
3560:,
3557:)
3554:x
3551:(
3548:f
3545:{
3537:=
3527:)
3524:x
3521:(
3516:+
3512:f
3477:d
3467:f
3461:E
3447:d
3441:+
3437:f
3431:E
3423:=
3417:d
3413:f
3408:E
3390:x
3386:x
3369:.
3363:+
3354:d
3349:|
3345:f
3341:|
3335:E
3317:x
3313:f
3309:f
3293:f
3275:t
3272:d
3268:)
3265:t
3262:(
3253:f
3242:0
3234:=
3231:f
3215:f
3211:}
3209:t
3205:x
3203:(
3201:f
3197:x
3193:μ
3189:t
3187:(
3185:f
3175:t
3171:x
3169:(
3167:f
3163:x
3159:μ
3149:t
3145:y
3139:t
3135:y
3130:t
3125:R
3121:R
3117:f
3112:f
3104:a
3100:b
3093:A
3088:)
3086:A
3084:(
3082:μ
3070:f
3064:b
3060:a
3054:f
2985:.
2975:|
2969:i
2960:)
2955:i
2951:t
2947:(
2944:f
2939:n
2934:1
2931:=
2928:i
2917:S
2913:|
2900:,
2868:]
2865:b
2862:,
2859:a
2856:[
2836:0
2810:0
2790:S
2786:f
2776:i
2773:Δ
2770:n
2766:i
2752:i
2748:x
2746:−
2743:i
2739:x
2734:i
2731:Δ
2714:;
2709:i
2700:)
2695:i
2691:t
2687:(
2684:f
2679:n
2674:1
2671:=
2668:i
2650:f
2639:i
2635:t
2630:i
2626:n
2607:.
2604:b
2601:=
2596:n
2592:x
2583:n
2579:t
2570:1
2564:n
2560:x
2545:2
2541:x
2532:2
2528:t
2519:1
2515:x
2506:1
2502:t
2493:0
2489:x
2485:=
2482:a
2412:x
2408:y
2391:x
2387:y
2354:x
2329:,
2324:3
2321:2
2316:=
2313:x
2310:d
2304:x
2297:1
2292:0
2255:,
2245:)
2239:5
2236:4
2226:5
2223:5
2217:(
2210:5
2207:5
2201:+
2195:+
2191:)
2185:5
2182:1
2172:5
2169:2
2163:(
2156:5
2153:2
2147:+
2143:)
2139:0
2131:5
2128:1
2122:(
2115:5
2112:1
2093:1
2072:0
2059:x
2052:x
2035:x
2022:x
2020:(
2018:f
1998:x
1968:g
1963:b
1958:a
1948:2
1944:c
1940:+
1937:f
1932:b
1927:a
1917:1
1913:c
1909:=
1906:)
1903:g
1898:2
1894:c
1890:+
1887:f
1882:1
1878:c
1874:(
1869:b
1864:a
1818:,
1815:x
1812:d
1808:)
1805:x
1802:(
1799:f
1775:b
1769:a
1764:)
1762:x
1760:(
1758:f
1752:x
1746:x
1731:∫
1714:.
1711:x
1708:d
1704:)
1701:x
1698:(
1695:f
1690:b
1685:a
1666:x
1661:)
1659:x
1657:(
1655:f
1627:′
1625:x
1616:x
1611:.
1579:(
1577:ſ
1573:∫
1469:x
1426:n
1416:n
1410:x
1376:k
1356:x
1353:d
1347:k
1343:x
1317:k
1313:x
1309:=
1306:y
1283:(
1213:(
1036:e
1029:t
1022:v
600:)
545:)
541:(
530:)
466:)
274:)
186:)
183:a
180:(
177:f
171:)
168:b
165:(
162:f
159:=
156:t
153:d
149:)
146:t
143:(
136:f
130:b
125:a
81:)
78:x
75:(
72:f
53:.
42:.
20:)
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