This article is about the concept of definite integrals in calculus. For the indefinite integral, seeantiderivative. For the set of numbers, seeinteger. For other uses, seeIntegral (disambiguation).
A definite integral of a function can be represented as thesigned area of the region bounded by its graph and the horizontal axis; in the above graph as an example, the integral of is the yellow (−) area subtracted from the blue (+) area
Inmathematics, anintegral is the continuous analog of asum, which is used to calculateareas,volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations ofcalculus,[a] the other beingdifferentiation. Integration was initially used to solve problems in mathematics andphysics, such as finding thearea under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.
Adefinite integral computes thesigned area of the region in the plane that is bounded by thegraph of a givenfunction between two points in thereal line. Conventionally, areas above the horizontalaxis of the plane are positive while areas below are negative. Integrals also refer to the concept of anantiderivative, a function whosederivative is the given function; in this case, they are also calledindefinite integrals. Thefundamental theorem of calculus 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 areinverse operations.
Although methods of calculating areas and volumes dated fromancient Greek mathematics, the principles of integration were formulated independently byIsaac Newton andGottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles ofinfinitesimal width.Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of acurvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century,Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as theLebesgue integral; it is more general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable.
Integrals may be generalized depending on the type of the function as well as thedomain over which the integration is performed. For example, aline integral is defined for functions of two or more variables, and theinterval of integration is replaced by a curve connecting two points in space. In asurface integral, the curve is replaced by a piece of asurface inthree-dimensional space.
The first documented systematic technique capable of determining integrals is themethod of exhaustion of theancient Greek astronomerEudoxus and philosopherDemocritus (ca. 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.[1] This method was further developed and employed byArchimedes in the 3rd century BC and used to calculate thearea of a circle, thesurface area andvolume of asphere, area of anellipse, the area under aparabola, the volume of a segment of aparaboloid of revolution, the volume of a segment of ahyperboloid of revolution, and the area of aspiral.[2]
A similar method was independently developed inChina around the 3rd century AD byLiu Hui, who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematiciansZu Chongzhi andZu Geng to find the volume of a sphere.[3]
In the Middle East, Hasan Ibn al-Haytham, Latinized asAlhazen (c. 965 – c. 1040 AD) derived a formula for the sum offourth powers.[4] Alhazen determined the equations to calculate the area enclosed by the curve represented by (which translates to the integral in contemporary notation), for any given non-negative integer value of.[5] 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 aparaboloid.[6]
Further steps were made in the early 17th century byBarrow andTorricelli, who provided the first hints of a connection between integration anddifferentiation. Barrow provided the first proof of thefundamental theorem of calculus.[9]Wallis generalized Cavalieri's method, computing integrals ofx to a general power, including negative powers and fractional powers.[10]
The major advance in integration came in the 17th century with the independent discovery of thefundamental theorem of calculus byLeibniz andNewton.[11] 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 moderncalculus, whose notation for integrals is drawn directly from the work of Leibniz.
While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree ofrigour.Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them "ghosts of departed quantities".[12] Calculus acquired a firmer footing with the development oflimits. Integration was first rigorously formalized, using limits, byRiemann.[13] Although all boundedpiecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context ofFourier analysis—to which Riemann's definition does not apply, andLebesgue formulated adifferent definition of integral, founded inmeasure theory (a subfield ofreal analysis). 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 thestandard part of an infinite Riemann sum, based on thehyperreal number system.
The notation for the indefinite integral was introduced byGottfried Wilhelm Leibniz in 1675.[14] He adapted theintegral symbol,∫, from the letterſ (long s), standing forsumma (written asſumma; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used byJoseph Fourier inMémoires of the French Academy around 1819–1820, reprinted in his book of 1822.[15]
Isaac Newton 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.x orx′, which are used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.[16]
In general, the integral of areal-valued functionf(x) with respect to a real variablex on an interval[a,b] is written as
The integral sign∫ represents integration. The symboldx, called thedifferential of the variablex, indicates that the variable of integration isx. The functionf(x) is called theintegrand, the pointsa andb are called the limits (or bounds) of integration, and the integral is said to be over the interval[a,b], called the interval of integration.[18] A function is said to beintegrable if its integral over its domain is finite. If limits are specified, the integral is called a definite integral.
When the limits are omitted, as in
the integral is called an indefinite integral, which represents a class of functions (theantiderivative) whose derivative is the integrand.[19] Thefundamental theorem of calculus 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).
In advanced settings, it is not uncommon to leave outdx when only the simpleRiemann integral is being used, or the exact type of integral is immaterial. For instance, one might write to express the linearity of the integral, a property shared by the Riemann integral and all generalizations thereof.[20]
Approximations to integral of√x from 0 to 1, with 5 yellow right endpoint partitions and 10 green left endpoint partitions
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, integrals are required to find exact and rigorous values for these quantities. In each case, one may divide the sought quantity into infinitely manyinfinitesimal pieces, then sum the pieces to achieve an accurate approximation.
As another example, to find the area of the region bounded by the graph of the functionf(x) = betweenx = 0 andx = 1, one can divide the interval into five pieces (0, 1/5, 2/5, ..., 1), then construct rectangles using the right end height of each piece (thus√0,√1/5,√2/5, ...,√1) and sum their areas to get the approximation
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, it will reach a limit which is the exact value of the area sought (in this case,2/3). One writes
which means2/3 is the result of a weighted sum of function values,√x, multiplied by the infinitesimal step widths, denoted bydx, on the interval[0, 1].
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 Riemann integrals and Lebesgue integrals.
The Riemann integral is defined in terms ofRiemann sums of functions with respect totagged partitions of an interval.[21] A tagged partition of aclosed interval[a,b] on the real line is a finite sequence
This partitions the interval[a,b] inton sub-intervals[xi−1,xi] indexed byi, each of which is "tagged" with a specific pointti ∈ [xi−1,xi]. ARiemann sum of a functionf with respect to such a tagged partition is defined as
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,Δi =xi−xi−1. Themesh of such a tagged partition is the width of the largest sub-interval formed by the partition,maxi=1...n Δi. TheRiemann integral of a functionf over the interval[a,b] is equal toS if:[22]
For all there exists such that, for any tagged partition with mesh less than,
When the chosen tags are the maximum (respectively, minimum) value of the function in each interval, the Riemann sum becomes an upper (respectively, lower)Darboux sum, suggesting the close connection between the Riemann integral and theDarboux integral.
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 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.[23]
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. ThusHenri Lebesgue introduced the integral bearing his name, explaining this integral thus in a letter toPaul Montel:[24]
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 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.
As Folland puts it, "To compute the Riemann integral off, one partitions the domain[a,b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range off ".[25] The definition of the Lebesgue integral thus begins with ameasure, μ. In the simplest case, theLebesgue measureμ(A) of an intervalA = [a,b] is its width,b −a, so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist.[26] In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.
Using the "partitioning the range off " philosophy, the integral of a non-negative functionf :R →R should be the sum overt of the areas between a thin horizontal strip betweeny =t andy =t +dt. This area is justμ{x :f(x) >t} dt. Letf∗(t) =μ{x :f(x) >t }. The Lebesgue integral off is then defined by
where the integral on the right is an ordinary improper Riemann integral (f∗ is a strictly decreasing positive function, and therefore has awell-defined improper Riemann integral).[27] For a suitable class of functions (themeasurable functions) this defines the Lebesgue integral.
A general measurable functionf is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph off and thex-axis is finite:[28]
In that case, the integral is, as in the Riemannian case, the difference between the area above thex-axis and the area below thex-axis:[29]
Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including:
TheDarboux integral, which is defined by Darboux sums (restricted Riemann sums) yet is equivalent to theRiemann integral. 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.
TheRiemann–Stieltjes integral, an extension of the Riemann integral which integrates with respect to a function as opposed to a variable.
Therough path integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against bothsemimartingales and processes such as thefractional Brownian motion.
TheChoquet integral, a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953.
The collection of Riemann-integrable functions on a closed interval[a,b] forms avector space under the operations ofpointwise addition and multiplication by a scalar, and the operation of integration
is alinear functional on this vector space. Thus, the collection of integrable functions is closed under takinglinear combinations, and the integral of a linear combination is the linear combination of the integrals:[30]
Similarly, the set ofreal-valued Lebesgue-integrable functions on a givenmeasure spaceE with measureμ is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral
is a linear functional on this vector space, so that:[29]
that is compatible with linear combinations.[31] In this situation, the linearity holds for the subspace of functions whose integral is an element ofV (i.e. "finite"). The most important special cases arise whenK isR,C, or a finite extension of the fieldQp ofp-adic numbers, andV is a finite-dimensional vector space overK, and whenK =C andV is a complexHilbert space.
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 ofDaniell for the case of real-valued functions on a setX, generalized byNicolas Bourbaki to functions with values in a locally compact topological vector space. SeeHildebrandt 1953 for an axiomatic characterization of the integral.
A number of general inequalities hold for Riemann-integrablefunctions defined on aclosed andboundedinterval[a,b] and can be generalized to other notions of integral (Lebesgue and Daniell).
Upper and lower bounds. An integrable functionf on[a,b], is necessarilybounded on that interval. Thus there arereal numbersm andM so thatm ≤f (x) ≤M for allx in[a,b]. Since the lower and upper sums off over[a,b] are therefore bounded by, respectively,m(b −a) andM(b −a), it follows that
Inequalities between functions.[32] Iff(x) ≤g(x) for eachx in[a,b] then each of the upper and lower sums off is bounded above by the upper and lower sums, respectively, ofg. Thus This is a generalization of the above inequalities, asM(b −a) is the integral of the constant function with valueM over[a,b]. In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, iff(x) <g(x) for eachx in[a,b], then
Subintervals. If[c,d] is a subinterval of[a,b] andf (x) is non-negative for allx, then
Products and absolute values of functions. Iff andg are two functions, then we may consider theirpointwise products and powers, andabsolute values: Iff is Riemann-integrable on[a,b] then the same is true for|f|, and Moreover, iff andg are both Riemann-integrable thenfg is also Riemann-integrable, and This inequality, known as theCauchy–Schwarz inequality, plays a prominent role inHilbert space theory, where the left hand side is interpreted as theinner product of twosquare-integrable functionsf andg on the interval[a,b].
Hölder's inequality.[33] Suppose thatp andq are two real numbers,1 ≤p,q ≤ ∞ with1/p +1/q = 1, andf andg are two Riemann-integrable functions. Then the functions|f|p and|g|q are also integrable and the followingHölder's inequality holds: Forp =q = 2, Hölder's inequality becomes the Cauchy–Schwarz inequality.
Minkowski inequality.[33] Suppose thatp ≥ 1 is a real number andf andg are Riemann-integrable functions. Then|f|p, |g|p and|f +g|p are also Riemann-integrable and the followingMinkowski inequality holds: An analogue of this inequality for Lebesgue integral is used in construction ofLp spaces.
over an interval[a,b] is defined ifa <b. This means that the upper and lower sums of the functionf are evaluated on a partitiona =x0 ≤x1 ≤ . . . ≤xn =b whose valuesxi are increasing. Geometrically, this signifies that integration takes place "left to right", evaluatingf within intervals[xi ,xi +1] where an interval with a higher index lies to the right of one with a lower index. The valuesa andb, the end-points of theinterval, are called thelimits of integration off. Integrals can also be defined ifa >b:[18]
Witha =b, this implies:
The first convention is necessary in consideration of taking integrals over subintervals of[a,b]; the second says that an integral taken over a degenerate interval, or apoint, should bezero. One reason for the first convention is that the integrability off on an interval[a,b] implies thatf is integrable on any subinterval[c,d], but in particular integrals have the property that ifc is anyelement of[a,b], then:[30]
With the first convention, the resulting relation
is then well-defined for any cyclic permutation ofa,b, andc.
Thefundamental theorem of calculus is the statement thatdifferentiation and integration are inverse operations: if acontinuous function is first integrated and then differentiated, the original function is retrieved.[34] An important consequence, sometimes called thesecond fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated.[35]
Letf be a real-valued function defined on aclosed interval [a,b] that admits anantiderivativeF on[a,b]. That is,f andF are functions such that for allx in[a,b],
Theimproper integral has unbounded intervals for both domain and range.
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 thelimit of asequence of properRiemann integrals on progressively larger intervals.
If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity:[37]
If the integrand is only defined or finite on a half-open interval, for instance(a,b], then again a limit may provide a finite result:[38]
That is, the improper integral is thelimit of proper integrals as one endpoint of the interval of integration approaches either a specifiedreal number, or∞, or−∞. In more complicated cases, limits are required at both endpoints, or at interior points.
Just as the definite integral of a positive function of one variable represents thearea of the region between the graph of the function and thex-axis, thedouble integral of a positive function of two variables represents thevolume of the region between the surface defined by the function and the plane that contains its domain.[39] For example, a function in two dimensions depends on two real variables,x andy, and the integral of a functionf over the rectangleR given as theCartesian product of two intervals can be written
where the differentialdA indicates that integration is taken with respect to area. Thisdouble integral can be defined usingRiemann sums, and represents the (signed) volume under the graph ofz =f(x,y) over the domainR.[40] Under suitable conditions (e.g., iff is continuous),Fubini's theorem states that this integral can be expressed as an equivalent iterated integral[41]
This reduces the problem of computing a double integral to computing one-dimensional integrals. Because of this, another notation for the integral overR uses a double integral sign:[40]
Integration over more general domains is possible. The integral of a functionf, with respect to volume, over ann-dimensional regionD of is denoted by symbols such as:
A line integral sums together elements along a curve.
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 withvector fields.
Aline integral (sometimes called apath integral) is an integral where thefunction to be integrated is evaluated along acurve.[42] Various different line integrals are in use. In the case of a closed curve it is also called acontour integral.
The function to be integrated may be ascalar field or avector field. 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 (commonlyarc length or, for a vector field, thescalar product of the vector field with adifferential vector in the curve).[43] This weighting distinguishes the line integral from simpler integrals defined onintervals. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact thatwork is equal toforce,F, multiplied by displacement,s, may be expressed (in terms of vector quantities) as:[44]
For an object moving along a pathC in avector fieldF such as anelectric field orgravitational field, the total work done by the field on the object is obtained by summing up the differential work done in moving froms tos +ds. This gives the line integral[45]
The definition of surface integral relies on splitting the surface into small surface elements.
Asurface integral generalizes double integrals to integration over asurface (which may be a curved set inspace); it can be thought of as thedouble integral analog of theline integral. The function to be integrated may be ascalar field or avector field. 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.[46]
For an example of applications of surface integrals, consider a vector fieldv on a surfaceS; that is, for each pointx inS,v(x) is a vector. Imagine that a fluid flows throughS, such thatv(x) determines the velocity of the fluid atx. Theflux is defined as the quantity of fluid flowing throughS in unit amount of time. To find the flux, one need to take thedot product ofv with the unitsurface normal toS at each point, which will give a scalar field, which is integrated over the surface:[47]
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 theclassical theory ofelectromagnetism.
Incomplex analysis, the integrand is acomplex-valued function of a complex variablez instead of a real function of a real variablex. When a complex function is integrated along a curve in the complex plane, the integral is denoted as follows
whereE,F,G 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 differentialsdx,dy,dz measure infinitesimal oriented lengths parallel to the three coordinate axes.
A differential two-form is a sum of the form
Here the basic two-forms measure oriented areas parallel to the coordinate two-planes. The symbol denotes thewedge product, which is similar to thecross product 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.
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). Theexterior derivative plays the role of thegradient andcurl of vector calculus, andStokes' theorem simultaneously generalizes the three theorems of vector calculus: thedivergence theorem,Green's theorem, and theKelvin-Stokes theorem.
An integration that is performed not over a variable (or, in physics, over a space or time dimension), but over aspace of functions, is referred to as afunctional integral.
Integrals are used extensively in many areas. For example, inprobability theory, integrals are used to determine the probability of somerandom variable falling within a certain range.[48] Moreover, the integral under an entireprobability density function must equal 1, which provides a test of whether afunction with no negative values could be a density function or not.[49]
Integrals can be used for computing thearea of a two-dimensional region that has a curved boundary, as well ascomputing the volume 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.[50] The volume of a three-dimensional object such as a disc or washer can be computed bydisc integration using the equation for the volume of a cylinder,, where is the radius. In the case of a simple disc created by rotating a curve about thex-axis, the radius is given byf(x), and its height is the differentialdx. Using an integral with boundsa andb, the volume of the disc is equal to:[51]Integrals are also used in physics, in areas likekinematics to find quantities likedisplacement,time, andvelocity. For example, inrectilinear motion, the displacement of an object over the time interval is given by
where is the velocity expressed as a function of time.[52] The work done by a force (given as a function of position) from an initial position to a final position is:[53]
The most basic technique for computing definite integrals of one real variable is based on thefundamental theorem of calculus. Letf(x) be the function ofx to be integrated over a given interval[a,b]. Then, find an antiderivative off; that is, a functionF such thatF′ =f on the interval. Provided the integrand and integral have nosingularities on the path of integration, by the fundamental theorem of calculus,
Alternative methods exist to compute more complex integrals. Manynonelementary integrals can be expanded in aTaylor series and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution usingMeijer G-functions 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,Parseval's identity 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, seeGaussian integral.
Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensivetables of integrals have been compiled and published over the years for this purpose. With the spread of computers, many professionals, educators, and students have turned tocomputer algebra systems 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, likeMacsyma andMaple.
A major mathematical difficulty in symbolic integration is that in many cases, a relatively simple function does not have integrals that can be expressed inclosed form involving onlyelementary functions, includerational andexponential functions,logarithm,trigonometric functions andinverse trigonometric functions, and the operations of multiplication and composition. TheRisch algorithm 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 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 inMathematica,Maple and othercomputer algebra systems, does just that for functions and antiderivatives built from rational functions,radicals, logarithm, and exponential functions.
Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, thespecial functions (like theLegendre functions, thehypergeometric function, thegamma function, theincomplete gamma function and so on). Extending Risch's algorithm to include such functions is possible but challenging and has been an active research subject.
More recently a new approach has emerged, usingD-finite functions, which are the solutions oflinear differential equations with polynomial coefficients. Most of the elementary and special functions areD-finite, and the integral of aD-finite function is also aD-finite function. This provides an algorithm to express the antiderivative of aD-finite function as the solution of a differential equation. This theory also allows one to compute the definite integral of aD-function as the sum of a series given by the first coefficients and provides an algorithm to compute any coefficient.
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.[54] The method of brackets 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 theMellin transform.[55]
Definite integrals may be approximated using several methods ofnumerical integration. Therectangle method 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, thetrapezoidal rule, 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.[56] The idea behind the trapezoidal rule, that more accurate approximations to the function yield better approximations to the integral, can be carried further:Simpson's rule approximates the integrand by a piecewise quadratic function.[57]
Riemann sums, the trapezoidal rule, and Simpson's rule are examples of a family of quadrature rules called theNewton–Cotes formulas. The degreen Newton–Cotes quadrature rule approximates the polynomial on each subinterval by a degreen polynomial. This polynomial is chosen to interpolate the values of the function on the interval.[58] Higher degree Newton–Cotes approximations can be more accurate, but they require more function evaluations, and they can suffer from numerical inaccuracy due toRunge's phenomenon. One solution to this problem isClenshaw–Curtis quadrature, in which the integrand is approximated by expanding it in terms ofChebyshev polynomials.
Romberg's method halves the step widths incrementally, giving trapezoid approximations denoted byT(h0),T(h1), and so on, wherehk+1 is half ofhk. For each new step size, only half the new function values need to be computed; the others carry over from the previous size. It theninterpolate a polynomial through the approximations, and extrapolate toT(0).Gaussian quadrature evaluates the function at the roots of a set oforthogonal polynomials.[59] Ann-point Gaussian method is exact for polynomials of degree up to2n − 1.
The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives asMonte Carlo integration.[60]
The area of an arbitrary two-dimensional shape can be determined using a measuring instrument calledplanimeter. The volume of irregular objects can be measured with precision by the fluiddisplaced as the object is submerged.
^Integral calculus is a very well established mathematical discipline for which there are many sources. SeeApostol 1967 andAnton, Bivens & Davis 2016, for example.
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