Integral
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral
Roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written:
A similar method was independently developed in China around the 3rd century AD by Liu Hui, who used it to find the area of the circle. This method was later used in the 5th century by Chinese fatherandson mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere (Shea 2007; Katz 2004, pp. 125–126).
In India around 15th century, in the Jyeṣṭhadeva veda, we find the notion of integration, termed sankalitam, (lit. collection), as in the statement:
The next significant advances in integral calculus did not begin to appear until the 17th century. At this time the work of Cavalieri with his method of Indivisibles, and work by Fermat, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of x^{n} up to degree n = 9 in Cavalieri's quadrature formula. Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers.
The modern notation for the indefinite integral was introduced by Gottfried Leibniz in 1675 (Burton 1988, p. 359; Leibniz 1899, p. 154). He adapted the integral symbol, ∫, from the letter ſ (long s), standing for summa (written as ſumma; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819–20, reprinted in his book of 1822 (Cajori 1929, pp. 249–250; Fourier 1822, §231).
When the limits are omitted, as in
To start off, consider the curve y = f(x) between x = 0 and x = 1 with f(x) = √x (see figure). We ask:
The notation
Historically, after the failure of early efforts to rigorously interpret infinitesimals, Riemann formally defined integrals as a limit of weighted sums, so that the dx suggested the limit of a difference (namely, the interval width). Shortcomings of Riemann's dependence on intervals and continuity motivated newer definitions, especially the Lebesgue integral, which is founded on an ability to extend the idea of "measure" in much more flexible ways. Thus the notation
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 Henri Lebesgue introduced the integral bearing his name, explaining this integral thus in a letter to Paul Montel:
Using the "partitioning the range of f" philosophy, the integral of a nonnegative function f : R → R should be the sum over t of the areas between a thin horizontal strip between y = t and y = t + dt. This area is just μ{ x : f(x) > t} dt. Let f^{∗}(t) = μ{ x : f(x) > t}. The Lebesgue integral of f is then defined by (Lieb & Loss 2001)
A general measurable function f is Lebesgueintegrable if the area between the graph of f and the xaxis is finite:
Linearity, together with some natural continuity properties and normalisation for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of Daniell for the case of realvalued functions on a set X, generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space. See (Hildebrandt 1953) for an axiomatic characterisation of the integral.
If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity.
A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use. In the case of a closed curve it is also called a contour integral.
The function to be integrated may be a scalar field or a vector 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 (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that work is equal to force, F, multiplied by displacement, s, may be expressed (in terms of vector quantities) as:
For an example of applications of surface integrals, consider a vector field v on a surface S; that is, for each point x in S, v(x) is a vector. Imagine that we have a fluid flowing through S, such that v(x) determines the velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through S in unit amount of time. To find the flux, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, which we integrate over the surface:
A differential twoform is a sum of the form
Unlike the cross product, and the threedimensional 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 higherdimensional analogs). The exterior derivative plays the role of the gradient and curl of vector calculus, and Stokes' theorem simultaneously generalizes the three theorems of vector calculus: the divergence theorem, Green's theorem, and the KelvinStokes theorem.
The most difficult step is usually to find the antiderivative of f. It is rarely possible to glance at a function and write down its antiderivative. More often, 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:
Computations of volumes of solids of revolution can usually be done with disk integration or shell integration.
Specific results which have been worked out by various techniques are collected in the list of integrals.
A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather simplelooking function does not exist. For instance, it is known that the antiderivatives of the functions exp(x^{2}), x^{x} and (sin x)/x cannot be expressed in the closed form involving only rational and exponential functions, logarithm, trigonometric and inverse trigonometric functions, and the operations of multiplication and composition; in other words, none of the three given functions is integrable in elementary functions, which are the functions which may be built from rational functions, roots of a polynomial, logarithm, and exponential functions. The Risch algorithm provides a general criterion to determine whether the antiderivative of an elementary function is elementary, and, if it is, to compute it. Unfortunately, it turns out that functions with closed expressions of antiderivatives are the exception rather than the rule. 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 be 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 Mathematica and other computer 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, the special functions of physics (like the Legendre functions, the hypergeometric function, the Gamma function, the Incomplete Gamma function and so on — see Symbolic integration for more details). Extending the 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, using Dfinite functions, which are the solutions of linear differential equations with polynomial coefficients. Most of the elementary and special functions are Dfinite, and the integral of a Dfinite function is also a Dfinite function. This provides an algorithm to express the antiderivative of a Dfinite function as the solution of a differential equation.
This theory also allows one to compute the definite integral of a Dfunction as the sum of a series given by the first coefficients, and provides an algorithm to compute any coefficient.^{[3]}
Consider, for example, the integral
A better approach replaces the rectangles used in a Riemann sum with trapezoids. The trapezoid rule is almost as easy to calculate; it sums all 17 function values, but weights the first and last by one half, and again multiplies by the step width. This immediately improves the approximation to 3.76925, which is noticeably more accurate. Furthermore, only 2^{10} pieces are needed to achieve 3.76000, substantially less computation than the rectangle method for comparable accuracy. The idea behind the trapezoid 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. Riemann sums, the trapezoid rule, and Simpson's rule are examples of a family of quadrature rules called Newton–Cotes formulas. The degree n Newton–Cotes quadrature rule approximates the polynomial on each subinterval by a degree n polynomial. This polynomial is chosen to interpolate the values of the function on the interval. Higher degree NewtonCotes approximations can be more accurate, but they require more function evaluations (already Simpson's rule requires twice the function evaluations of the trapezoid rule), and they can suffer from numerical inaccuracy due to Runge's phenomenon. One solution to this problem is Clenshaw–Curtis quadrature. In Clenshaw–Curtis quadrature, the integrand is approximated by expanding it in terms of Chebyshev polynomials. This produces an approximation whose values never deviate far from those of the original function.
Romberg's method builds on the trapezoid method to great effect. First, the step lengths are halved incrementally, giving trapezoid approximations denoted by T(h_{0}), T(h_{1}), and so on, where h_{k+1} is half of h_{k}. For each new step size, only half the new function values need to be computed; the others carry over from the previous size (as shown in the table above). But the really powerful idea is to interpolate a polynomial through the approximations, and extrapolate to T(0). With this method a numerically exact answer here requires only four pieces (five function values). The Lagrange polynomial interpolating {h_{k},T(h_{k})}_{k = 0...2} = {(4.00,6.128), (2.00,4.352), (1.00,3.908)} is 3.76 + 0.148h^{2}, producing the extrapolated value 3.76 at h = 0.
Gaussian quadrature often requires noticeably less work for superior accuracy. In this example, it can compute the function values at just two x positions, ±2 ⁄ √3, then double each value and sum to get the numerically exact answer. The explanation for this dramatic success lies in the choice of points. Unlike Newton–Cotes rules, which interpolate the integrand at evenly spaced points, Gaussian quadrature evaluates the function at the roots of a set of orthogonal polynomials. An npoint Gaussian method is exact for polynomials of degree up to 2n − 1. The function in this example is a degree 3 polynomial, plus a term that cancels because the chosen endpoints are symmetric around zero. (Cancellation also benefits the Romberg method.)
In practice, each method must use extra evaluations to ensure an error bound on an unknown function; this tends to offset some of the advantage of the pure Gaussian method, and motivates the popular Gauss–Kronrod quadrature formulae. More broadly, adaptive quadrature partitions a range into pieces based on function properties, so that data points are concentrated where they are needed most.
The computation of higherdimensional integrals (for example, volume calculations) makes important use of such alternatives as Monte Carlo integration.
A calculus text is no substitute for numerical analysis, but the reverse is also true. Even the best adaptive numerical code sometimes requires a user to help with the more demanding integrals. For example, improper integrals may require a change of variable or methods that can avoid infinite function values, and known properties like symmetry and periodicity may provide critical leverage. For example, the integral is difficult to evaluate numerically because it is infinite at x = 0. However, the substitution u = √x transforms the integral into , which has no singularities at all.
Roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written:
Contents
History[edit]
See also: History of calculus
Precalculus integration[edit]
The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus (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. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle.A similar method was independently developed in China around the 3rd century AD by Liu Hui, who used it to find the area of the circle. This method was later used in the 5th century by Chinese fatherandson mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere (Shea 2007; Katz 2004, pp. 125–126).
In India around 15th century, in the Jyeṣṭhadeva veda, we find the notion of integration, termed sankalitam, (lit. collection), as in the statement:
 ekadyekothara pada sankalitam samam padavargathinte pakuti,
The next significant advances in integral calculus did not begin to appear until the 17th century. At this time the work of Cavalieri with his method of Indivisibles, and work by Fermat, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of x^{n} up to degree n = 9 in Cavalieri's quadrature formula. Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers.
Newton and Leibniz[edit]
The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz. 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 Newton and Leibniz developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern calculus, whose notation for integrals is drawn directly from the work of Leibniz.Formalization[edit]
While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them "ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits. Integration was first rigorously formalized, using limits, by Riemann. Although all bounded piecewise continuous functions are Riemannintegrable on a bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis—to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield of real 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 the standard part of an infinite Riemann sum, based on the hyperreal number system.Historical notation[edit]
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 or x′, which Newton used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.The modern notation for the indefinite integral was introduced by Gottfried Leibniz in 1675 (Burton 1988, p. 359; Leibniz 1899, p. 154). He adapted the integral symbol, ∫, from the letter ſ (long s), standing for summa (written as ſumma; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819–20, reprinted in his book of 1822 (Cajori 1929, pp. 249–250; Fourier 1822, §231).
Terminology and notation[edit]
Standard[edit]
The integral with respect to x of a realvalued function f(x) of a real variable x on the interval [a, b] is written as .
When the limits are omitted, as in
Variants[edit]
In modern Arabic mathematical notation, a reflected integral symbol is used instead of the symbol ∫.^{[1]} Some authors use an upright "d" to indicate the variable of integration (i.e., dx instead of dx). The symbol dx is not always placed after f(x), as for instance in .
Interpretations of the integral[edit]
Integrals appear in many practical situations. If a swimming pool is rectangular with a flat bottom, then from its length, width, and depth we can easily determine the volume of water it can contain (to fill it), the area of its surface (to cover it), and the length of its edge (to rope it). But if it is oval with a rounded bottom, all of these quantities call for integrals. Practical approximations may suffice for such trivial examples, but precision engineering (of any discipline) requires exact and rigorous values for these elements.To start off, consider the curve y = f(x) between x = 0 and x = 1 with f(x) = √x (see figure). We ask:
 What is the area under the function f, in the interval from 0 to 1?
The notation
Historically, after the failure of early efforts to rigorously interpret infinitesimals, Riemann formally defined integrals as a limit of weighted sums, so that the dx suggested the limit of a difference (namely, the interval width). Shortcomings of Riemann's dependence on intervals and continuity motivated newer definitions, especially the Lebesgue integral, which is founded on an ability to extend the idea of "measure" in much more flexible ways. Thus the notation
Formal definitions[edit]
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 also occasionally for pedagogical reasons. The most commonly used definitions of integral are Riemann integrals and Lebesgue integrals.Riemann integral[edit]
Main article: Riemann integral
The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval.^{[2]} Let [a, b] be a closed interval of the real line; then a tagged partition of [a, b] is a finite sequence For all ε > 0 there exists δ > 0 such that, for any tagged partition [a, b] with mesh less than δ, we have
Lebesgue integral[edit]
Main article: Lebesgue integration
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 Riemannintegrable, 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 (Rudin 1987).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 Henri Lebesgue introduced the integral bearing his name, explaining this integral thus in a letter to Paul Montel:
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 (1984, p. 56) puts it, "To compute the Riemann integral of f, one partitions the domain [a, b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f". The definition of the Lebesgue integral thus begins with a measure, μ. In the simplest case, the Lebesgue measure μ(A) of an interval A = [a, b] is its width, b − a, 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.
Using the "partitioning the range of f" philosophy, the integral of a nonnegative function f : R → R should be the sum over t of the areas between a thin horizontal strip between y = t and y = t + dt. This area is just μ{ x : f(x) > t} dt. Let f^{∗}(t) = μ{ x : f(x) > t}. The Lebesgue integral of f is then defined by (Lieb & Loss 2001)
A general measurable function f is Lebesgueintegrable if the area between the graph of f and the xaxis is finite:
Other integrals[edit]
Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including: The Darboux integral, which is constructed using Darboux sums and is equivalent to a Riemann integral, meaning that a function is Darbouxintegrable if and only if it is Riemannintegrable. Darboux integrals have the advantage of being simpler to define than Riemann integrals.
 The Riemann–Stieltjes integral, an extension of the Riemann integral.
 The Lebesgue–Stieltjes integral, further developed by Johann Radon, which generalizes the Riemann–Stieltjes and Lebesgue integrals.
 The Daniell integral, which subsumes the Lebesgue integral and Lebesgue–Stieltjes integral without the dependence on measures.
 The Haar integral, used for integration on locally compact topological groups, introduced by Alfréd Haar in 1933.
 The Henstock–Kurzweil integral, variously defined by Arnaud Denjoy, Oskar Perron, and (most elegantly, as the gauge integral) Jaroslav Kurzweil, and developed by Ralph Henstock.
 The Itō integral and Stratonovich integral, which define integration with respect to semimartingales such as Brownian motion.
 The Young integral, which is a kind of Riemann–Stieltjes integral with respect to certain functions of unbounded variation.
 The rough path integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both semimartingales and processes such as the fractional Brownian motion.
Properties[edit]
Linearity[edit]
The collection of Riemannintegrable functions on a closed interval [a, b] forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integrationLinearity, together with some natural continuity properties and normalisation for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of Daniell for the case of realvalued functions on a set X, generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space. See (Hildebrandt 1953) for an axiomatic characterisation of the integral.
Inequalities[edit]
A number of general inequalities hold for Riemannintegrable functions defined on a closed and bounded interval [a, b] and can be generalized to other notions of integral (Lebesgue and Daniell). Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that m ≤ f (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(b − a) and M(b − a), it follows that
 Inequalities between functions. If f(x) ≤ g(x) for each x in [a, b] then each of the upper and lower sums of f is bounded above by the upper and lower sums, respectively, of g. Thus

 This is a generalization of the above inequalities, as M(b − a) is the integral of the constant function with value M over [a, b].
 In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if f(x) < g(x) for each x in [a, b], then
 Subintervals. If [c, d] is a subinterval of [a, b] and f(x) is nonnegative for all x, then
 Products and absolute values of functions. If f and g are two functions then we may consider their pointwise products and powers, and absolute values:

 If f is Riemannintegrable on [a, b] then the same is true for f, and
 Moreover, if f and g are both Riemannintegrable then fg is also Riemannintegrable, and
 This inequality, known as the Cauchy–Schwarz inequality, plays a prominent role in Hilbert space theory, where the left hand side is interpreted as the inner product of two squareintegrable functions f and g on the interval [a, b].
 Hölder's inequality. Suppose that p and q are two real numbers, 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, and f and g are two Riemannintegrable functions. Then the functions f^{p} and g^{q} are also integrable and the following Hölder's inequality holds:
 For p = q = 2, Hölder's inequality becomes the Cauchy–Schwarz inequality.
 Minkowski inequality. Suppose that p ≥ 1 is a real number and f and g are Riemannintegrable functions. Then  f ^{p},  g ^{p} and  f + g ^{p} are also Riemannintegrable and the following Minkowski inequality holds:
 An analogue of this inequality for Lebesgue integral is used in construction of L^{p} spaces.
Conventions[edit]
In this section f is a realvalued Riemannintegrable function. The integral Reversing limits of integration. If a > b then define
 Integrals over intervals of length zero. If a is a real number then
 Additivity of integration on intervals. If c is any element of [a, b], then
Fundamental theorem of calculus[edit]
Main article: Fundamental theorem of calculus
The fundamental theorem of calculus is the statement that differentiation and integration are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieved. An important consequence, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated.Statements of theorems[edit]
Fundamental theorem of calculus[edit]
Let f be a continuous realvalued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], bySecond fundamental theorem of calculus[edit]
Let f be a realvalued function defined on a closed interval [a, b] that admits an antiderivative F on [a, b]. That is, f and F are functions such that for all x in [a, b],Calculating integrals[edit]
The second fundamental theorem allows many integrals to be calculated explicitly. For example, to calculate the integralExtensions[edit]
Improper integrals[edit]
Main article: Improper integral
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 limit of a sequence of proper Riemann 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.
Multiple integration[edit]
Main article: Multiple integral
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the xaxis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function and the plane which contains its domain. For example, a function in two dimensions depends on two real variables, x and y, and the integral of a function f over the rectangle R given as the Cartesian product of two intervals can be writtenLine integrals[edit]
Main article: Line integral
The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces. Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with vector fields.A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use. In the case of a closed curve it is also called a contour integral.
The function to be integrated may be a scalar field or a vector 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 (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that work is equal to force, F, multiplied by displacement, s, may be expressed (in terms of vector quantities) as:
Surface integrals[edit]
Main article: Surface integral
A surface integral is a definite integral taken over a surface (which may be a curved set in space); it can be thought of as the double integral analog of the line integral. The function to be integrated may be a scalar field or a vector 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.For an example of applications of surface integrals, consider a vector field v on a surface S; that is, for each point x in S, v(x) is a vector. Imagine that we have a fluid flowing through S, such that v(x) determines the velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through S in unit amount of time. To find the flux, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, which we integrate over the surface:
Contour integrals[edit]
In complex analysis, the integrand is a complexvalued function of a complex variable z instead of a real function of a real variable x. When a complex function is integrated along a curve in the complex plane, the integral is denoted as follows .
Integrals of differential forms[edit]
Main article: differential form
A differential form is a mathematical concept in the fields of multivariable calculus, differential topology, and tensors. Differential forms are organized by degree. For example, a oneform is a weighted sum of the differentials of the coordinates, such as:A differential twoform is a sum of the form
Unlike the cross product, and the threedimensional 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 higherdimensional analogs). The exterior derivative plays the role of the gradient and curl of vector calculus, and Stokes' theorem simultaneously generalizes the three theorems of vector calculus: the divergence theorem, Green's theorem, and the KelvinStokes theorem.
Summations[edit]
The discrete equivalent of integration is summation. Summations and integrals can be put on the same foundations using the theory of Lebesgue integrals or time scale calculus.Computation[edit]
Analytical[edit]
The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus. Let f(x) be the function of x to be integrated over a given interval [a, b]. Then, find an antiderivative of f; that is, a function F such that F′ = f on the interval. Provided the integrand and integral have no singularities on the path of integration, by the fundamental theorem of calculus,The most difficult step is usually to find the antiderivative of f. It is rarely possible to glance at a function and write down its antiderivative. More often, 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:
 Integration by substitution
 Integration by parts
 Inverse function integration
 Changing the order of integration
 Integration by trigonometric substitution
 Tangent halfangle substitution
 Integration by partial fractions
 Integration by reduction formulae
 Integration using parametric derivatives
 Integration using Euler's formula
 Euler substitution
 Differentiation under the integral sign
 Contour integration
Computations of volumes of solids of revolution can usually be done with disk integration or shell integration.
Specific results which have been worked out by various techniques are collected in the list of integrals.
Symbolic[edit]
Main article: Symbolic integration
Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive tables 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 to computer 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, like Macsyma.A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather simplelooking function does not exist. For instance, it is known that the antiderivatives of the functions exp(x^{2}), x^{x} and (sin x)/x cannot be expressed in the closed form involving only rational and exponential functions, logarithm, trigonometric and inverse trigonometric functions, and the operations of multiplication and composition; in other words, none of the three given functions is integrable in elementary functions, which are the functions which may be built from rational functions, roots of a polynomial, logarithm, and exponential functions. The Risch algorithm provides a general criterion to determine whether the antiderivative of an elementary function is elementary, and, if it is, to compute it. Unfortunately, it turns out that functions with closed expressions of antiderivatives are the exception rather than the rule. 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 be 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 Mathematica and other computer 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, the special functions of physics (like the Legendre functions, the hypergeometric function, the Gamma function, the Incomplete Gamma function and so on — see Symbolic integration for more details). Extending the 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, using Dfinite functions, which are the solutions of linear differential equations with polynomial coefficients. Most of the elementary and special functions are Dfinite, and the integral of a Dfinite function is also a Dfinite function. This provides an algorithm to express the antiderivative of a Dfinite function as the solution of a differential equation.
This theory also allows one to compute the definite integral of a Dfunction as the sum of a series given by the first coefficients, and provides an algorithm to compute any coefficient.^{[3]}
Numerical[edit]
Main article: Numerical integration
Some integrals found in real applications can be computed by closedform antiderivatives. Others are not so accommodating. Some antiderivatives do not have closed forms, some closed forms require special functions which themselves are a challenge to compute, and others are so complex that finding the exact answer is too slow. This motivates the study and application of numerical approximations of integrals. This subject, called numerical integration or numerical quadrature, arose early in the study of integration for the purpose of making hand calculations. The development of generalpurpose computers made numerical integration more practical and drove a desire for improvements. The goals of numerical integration are accuracy, reliability, efficiency, and generality, and sophisticated modern methods can vastly outperform a naive method by all four measures (Dahlquist & Björck 2008; Kahaner, Moler & Nash 1989; Stoer & Bulirsch 2002).Consider, for example, the integral
Spaced function values x −2.00 −1.50 −1.00 −0.50 0.00 0.50 1.00 1.50 2.00 f(x) 2.22800 2.45663 2.67200 2.32475 0.64400 −0.92575 −0.94000 −0.16963 0.83600 x −1.75 −1.25 −0.75 −0.25 0.25 0.75 1.25 1.75 f(x) 2.33041 2.58562 2.62934 1.64019 −0.32444 −1.09159 −0.60387 0.31734
A better approach replaces the rectangles used in a Riemann sum with trapezoids. The trapezoid rule is almost as easy to calculate; it sums all 17 function values, but weights the first and last by one half, and again multiplies by the step width. This immediately improves the approximation to 3.76925, which is noticeably more accurate. Furthermore, only 2^{10} pieces are needed to achieve 3.76000, substantially less computation than the rectangle method for comparable accuracy. The idea behind the trapezoid 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. Riemann sums, the trapezoid rule, and Simpson's rule are examples of a family of quadrature rules called Newton–Cotes formulas. The degree n Newton–Cotes quadrature rule approximates the polynomial on each subinterval by a degree n polynomial. This polynomial is chosen to interpolate the values of the function on the interval. Higher degree NewtonCotes approximations can be more accurate, but they require more function evaluations (already Simpson's rule requires twice the function evaluations of the trapezoid rule), and they can suffer from numerical inaccuracy due to Runge's phenomenon. One solution to this problem is Clenshaw–Curtis quadrature. In Clenshaw–Curtis quadrature, the integrand is approximated by expanding it in terms of Chebyshev polynomials. This produces an approximation whose values never deviate far from those of the original function.
Romberg's method builds on the trapezoid method to great effect. First, the step lengths are halved incrementally, giving trapezoid approximations denoted by T(h_{0}), T(h_{1}), and so on, where h_{k+1} is half of h_{k}. For each new step size, only half the new function values need to be computed; the others carry over from the previous size (as shown in the table above). But the really powerful idea is to interpolate a polynomial through the approximations, and extrapolate to T(0). With this method a numerically exact answer here requires only four pieces (five function values). The Lagrange polynomial interpolating {h_{k},T(h_{k})}_{k = 0...2} = {(4.00,6.128), (2.00,4.352), (1.00,3.908)} is 3.76 + 0.148h^{2}, producing the extrapolated value 3.76 at h = 0.
Gaussian quadrature often requires noticeably less work for superior accuracy. In this example, it can compute the function values at just two x positions, ±2 ⁄ √3, then double each value and sum to get the numerically exact answer. The explanation for this dramatic success lies in the choice of points. Unlike Newton–Cotes rules, which interpolate the integrand at evenly spaced points, Gaussian quadrature evaluates the function at the roots of a set of orthogonal polynomials. An npoint Gaussian method is exact for polynomials of degree up to 2n − 1. The function in this example is a degree 3 polynomial, plus a term that cancels because the chosen endpoints are symmetric around zero. (Cancellation also benefits the Romberg method.)
In practice, each method must use extra evaluations to ensure an error bound on an unknown function; this tends to offset some of the advantage of the pure Gaussian method, and motivates the popular Gauss–Kronrod quadrature formulae. More broadly, adaptive quadrature partitions a range into pieces based on function properties, so that data points are concentrated where they are needed most.
The computation of higherdimensional integrals (for example, volume calculations) makes important use of such alternatives as Monte Carlo integration.
A calculus text is no substitute for numerical analysis, but the reverse is also true. Even the best adaptive numerical code sometimes requires a user to help with the more demanding integrals. For example, improper integrals may require a change of variable or methods that can avoid infinite function values, and known properties like symmetry and periodicity may provide critical leverage. For example, the integral is difficult to evaluate numerically because it is infinite at x = 0. However, the substitution u = √x transforms the integral into , which has no singularities at all.
Mechanical[edit]
The area of an arbitrary twodimensional shape can be determined using a measuring instrument called planimeter. The volume of irregular objects can be measured with precision by the fluid displaced as the object is submerged.Geometrical[edit]
Main article: Quadrature (mathematics)
Area can be found via geometrical compassandstraightedge constructions of an equivalent square, e.g., squaring the circle._{}^{}
Integral
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