Integral

The word "integral" (adjective) can also mean: "being an integer". The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

This article is about the concept of integrals in calculus. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives For other meanings, see integration and integral (disambiguation).

A definite integral of a function can be represented as the signed area of the region bounded by its graph.

Integration is a core concept of advanced mathematics, specifically in the fields of calculus and mathematical analysis. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Analysis has its beginnings in the rigorous formulation of Calculus. Given a function f(x) of a real variable x and an interval [a,b] of the real line, the integral

$\int_a^b f(x)\,dx$

is equal to the area of a region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b, with areas below the x-axis being subtracted. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the real numbers may be described informally in several different ways A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x)

The term "integral" may also refer to the notion of antiderivative, a function F whose derivative is the given function f. In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In this case it is called an indefinite integral, while the integrals discussed in this article are termed definite integrals. Some authors maintain a distinction between antiderivatives and indefinite integrals.

The principles of integration were formulated by Isaac Newton and Gottfried Leibniz in the late seventeenth century. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if f is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by:

$\int_a^b f(x)\,dx = F(b) - F(a).$

Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" Beginning in the nineteenth century, more sophisticated notions of integral began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral is defined for functions of two or three variables, and the interval of integration [a,b] is replaced by a certain curve connecting two points on the plane or in the space. In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. In Mathematics, a surface integral is a Definite integral taken over a Surface (which may be a curved set in Space) it can be thought In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. Integrals of differential forms play a fundamental role in modern differential geometry. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry These generalizations of integral first arose from the needs of physics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently Modern concepts of integration are based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue. In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of Henri Léon Lebesgue leɔ̃ ləˈbɛg ( June 28, 1875, Beauvais &ndash July 26, 1941, Paris) was a French

History

See also: History of calculus

Pre-calculus integration

Integration can be traced as far back as ancient Egypt, circa 1800 BC, with the Moscow Mathematical Papyrus demonstrating knowledge of a formula for the volume of a pyramidal frustum. This is a sub-article to Calculus and History of mathematics. The Moscow Mathematical Papyrus is also called the Golenischev Mathematical Papyrus, after its first owner Egyptologist Vladimir Goleniščev. A pyramid is a Building where the upper surfaces are triangular and converge on one point Elements special cases and related concepts Each plane section is a base of the frustum The first documented systematic technique capable of determining integrals is the method of exhaustion of Eudoxus (circa 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known. The method of exhaustion is a method of finding the Area of a Shape by inscribing inside it a sequence of Polygons whose areas converge to the Eudoxus of Cnidus ( Greek Εὔδοξος ὁ Κνίδιος (410 or 408 BC &ndash 355 or 347 BC was a Greek Astronomer, Mathematician This method was further developed and employed by Archimedes and used to calculate areas for parabolas and an approximation to the area of a circle. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer Similar methods were independently developed in China around the 3rd Century AD by Liu Hui, who used it to find the area of the circle. Liu Hui ( fl 3rd century) was a Chinese Mathematician who lived in the Wei Kingdom. This method was later used by Zu Chongzhi to find the volume of a sphere. Zu Chongzhi ( 429–500 Courtesy name Wenyuan (文遠 was a prominent Chinese mathematician and astronomer during the Liu Some ideas of integral calculus are found in the Siddhanta Shiromani, a 12th century astronomy text by Indian mathematician Bhāskara II. Bhaskara (1114 &ndash 1185 also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher" was an Indian mathematician

Significant advances on techniques such as the method of exhaustion did not begin to appear until the 16th century AD. At this time the work of Cavalieri with his method of indivisibles, and work by Fermat, began to lay the foundations of modern calculus. Bonaventura Francesco Cavalieri (in Latin, Cavalerius) ( 1598 - November 30, 1647) was an Italian mathematician Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the 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. Isaac Barrow (October 1630 &ndash May 4, 1677) was an English scholar and Mathematician who is generally given credit for his early role Evangelista Torricelli ( ( October 15, 1608 &ndash October 25, 1647) was an Italian physicist and mathematician Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change

Newton and Leibniz

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 fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements 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. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

Formalizing integrals

While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigor. Bishop Berkeley memorably attacked infinitesimals as "the ghosts of departed quantity". George Berkeley (ˈbɑrkli (12 March 1685 14 January 1753 also known as Bishop Berkeley, was a Philosopher. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have Calculus acquired a firmer footing with the development of limits and was given a suitable foundation by Cauchy in the first half of the 19th century. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" Integration was first rigorously formalized, using limits, by Riemann. Although all bounded piecewise continuous functions are Riemann integrable on a bounded interval, subsequently more general functions were considered, to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory. Henri Léon Lebesgue leɔ̃ ləˈbɛg ( June 28, 1875, Beauvais &ndash July 26, 1941, Paris) was a French In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed.

Notation

Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements The vertical bar was easily confused with $\dot{x}$ 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 an elongated letter S, standing for summa (Latin for "sum" or "total"). The long, medial or descending s ( ſ) is a form of the minuscule letter ' S ' formerly used where 's' occurred in the middle 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. Jean Baptiste Joseph Fourier ( March 21, 1768 &ndash May 16, 1830) was a French Mathematician and Physicist 249–250; Fourier 1822, §231). In Arabic mathematical notation which is written from right to left, an inverted integral symbol is used (W3C 2006). The designation modern Arabic mathematical notation is used for a Mathematical notation based on the Arabic script that is widely used in the Arab world

Terminology and notation

If a function has an integral, it is said to be integrable. The function for which the integral is calculated is called the integrand. The region over which a function is being integrated is called the domain of integration. If the integral does not have a domain of integration, it is considered indefinite (one with a domain is considered definite). In general, the integrand may be a function of more than one variable, and the domain of integration may be an area, volume, a higher dimensional region, or even an abstract space that does not have a geometric structure in any usual sense.

The simplest case, the integral of a real-valued function f of one real variable x on the interval [a, b], is denoted by

$\int_a^b f(x)\,dx .$

The ∫ sign, an elongated "S", represents integration; a and b are the lower limit and upper limit of integration, defining the domain of integration; f is the integrand, to be evaluated as x varies over the interval [a,b]; and dx can have different interpretations depending on the theory being used. For example, it can be seen as merely a notation indicating that x is the 'dummy variable' of integration, as a reflection of the weights in the Riemann sum, a measure (in Lebesgue integration and its extensions), an infinitesimal (in non-standard analysis) or as an independent mathematical quantity: a differential form. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is More complicated cases may vary the notation slightly.

Introduction

Integrals appear in many practical situations. Consider a swimming pool. If it is rectangular, 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 at first, but eventually we demand exact and rigorous answers to such problems.

Approximations to integral of √x from 0 to 1, with ■ 5 right samples (above) and ■ 12 left samples (below)

To start off, consider the curve y = f(x) between x = 0 and x = 1, with f(x) = √x. We ask:

What is the area under the function f, in the interval from 0 to 1?

and call this (yet unknown) area the integral of f. The notation for this integral will be

$\int_0^1 \sqrt x \, dx \,\!.$

As a first approximation, look at the unit square given by the sides x = 0 to x = 1 and y = f(0) = 0 and y = f(1) = 1. Its area is exactly 1. As it is, the true value of the integral must be somewhat less. Decreasing the width of the approximation rectangles shall give a better result; so cross the interval in five steps, using the approximation points 0, ¹⁄₅, ²⁄₅, and so on to 1. Fit a box for each step using the right end height of each curve piece, thus √¹⁄₅, √²⁄₅, and so on to √1 = 1. Summing the areas of these rectangles, we get a better approximation for the sought integral, namely

$\textstyle \sqrt {\frac {1} {5}} \left ( \frac {1} {5} - 0 \right ) + \sqrt {\frac {2} {5}} \left ( \frac {2} {5} - \frac {1} {5} \right ) + \cdots + \sqrt {\frac {5} {5}} \left ( \frac {5} {5} - \frac {4} {5} \right ) \approx 0.7497\,\!$

Notice that we are taking a sum of finitely many function values of f, multiplied with the differences of two subsequent approximation points. We can easily see that the approximation is still too large. Using more steps produces a closer approximation, but will never be exact: replacing the 5 subintervals by twelve as depicted, we will get an approximate value for the area of 0. 6203, which is too small. The key idea is the transition from adding finitely many differences of approximation points multiplied by their respective function values to using infinitely fine, or infinitesimal steps. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have

As for the actual calculation of integrals, the fundamental theorem of calculus, due to Newton and Leibniz, is the fundamental link between the operations of differentiating and integrating. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change Applied to the square root curve, f(x) = x^1/2, it says to look at the related function F(x) = ²⁄₃x^3/2, and simply take F(1) − F(0), where 0 and 1 are the boundaries of the interval [0,1]. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set (This is a case of a general rule, that for f(x) = x^q, with q ≠ −1, the related function, the so-called antiderivative is F(x) = (x^q+1)/(q + 1). In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative ) So the exact value of the area under the curve is computed formally as

$\int_0^1 \sqrt x \cdot dx = \int_0^1 x^{1/2} \cdot dx = \int_0^1 d \left({\textstyle \frac 2 3} x^{3/2}\right) = {\textstyle \frac 2 3}.$

The notation

$\int f(x) \, dx \,\!$

conceives the integral as a weighted sum, denoted by the elongated "S", with function values, f(x), multiplied by infinitesimal step widths, the so-called differentials, denoted by dx. The multiplication sign is usually omitted.

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). In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" 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. In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of Thus the notation

$\int_A f(x) \, d\mu \,\!$

refers to a weighted sum in which the function values are partitioned, with μ measuring the weight to be assigned to each value. Here A denotes the region of integration.

Differential geometry, with its "calculus on manifolds", gives the familiar notation yet another interpretation. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Now f(x) and dx become a differential form, ω = f(x) dx, a new differential operator d, known as the exterior derivative appears, and the fundamental theorem becomes the more general Stokes' theorem,

$\int_{A} \bold{d} \omega = \int_{\part A} \omega , \,\!$

from which Green's theorem, the divergence theorem, and the fundamental theorem of calculus follow. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from In Physics and Mathematics, Green's theorem gives the relationship between a Line integral around a simple closed curve C and a Double integral In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration.

More recently, infinitesimals have reappeared with rigor, through modern innovations such as non-standard analysis. Non-standard analysis is a branch of Mathematics that formulates analysis using a rigorous notion of an Infinitesimal number Not only do these methods vindicate the intuitions of the pioneers, they also lead to new mathematics.

Although there are differences between these conceptions of integral, there is considerable overlap. Thus the area of the surface of the oval swimming pool can be handled as a geometric ellipse, as a sum of infinitesimals, as a Riemann integral, as a Lebesgue integral, or as a manifold with a differential form. The calculated result will be the same for all.

Formal definitions

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

Main article: Riemann integral

Integral approached as Riemann sum based on tagged partition, with irregular sampling positions and widths (max in red). In the branch of Mathematics known as Real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the Integral True value is 3. 76; estimate is 3. 648.

The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. In Mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph otherwise known as an Integral. Let [a,b] be a closed interval of the real line; then a tagged partition of [a,b] is a finite sequence

$a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!$

Riemann sums converging as intervals halve, whether sampled at ■ right, ■ minimum, ■ maximum, or ■ left. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set

This partitions the interval [a,b] into i sub-intervals [x_i−1, x_i], each of which is "tagged" with a distinguished point t_i ∈ [x_i−1, x_i]. Let Δ_i = x_i−x_i−1 be the width of sub-interval i; then the mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, max_i=1…n Δ_i. A Riemann sum of a function f with respect to such a tagged partition is defined as

$\sum_{i=1}^{n} f(t_i) \Delta_i ;$

thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. The Riemann integral of a function f over the interval [a,b] is equal to S if:

For all ε > 0 there exists δ > 0 such that, for any tagged partition [a,b] with mesh less than δ, we have

$\left| S - \sum_{i=1}^{n} f(t_i)\Delta_i \right| < \epsilon.$

When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the Darboux integral. If you are having difficulty understanding this article you might wish to learn more about Algebra, functions and mathematical limits. If you are having difficulty understanding this article you might wish to learn more about Algebra, functions and mathematical limits.

Lebesgue integral

Main article: Lebesgue integration

The Riemann integral is not defined for a wide range of functions and situations of importance in applications (and of interest in theory). In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics, matrix calculus is a specialized notation for doing Multivariable calculus, especially over spaces of matrices, where it defines the In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable In Calculus, the quotient rule is a method of finding the Derivative of a function that is the Quotient of two other functions for which In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. In Mathematics, an implicit function is a generalization for the concept of a function in which the Dependent variable has not been given "explicitly" In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor In Differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change The primary operation in Differential calculus is finding a Derivative. See the following pages for lists of Integrals: List of integrals of rational functions List of integrals of irrational functions In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other Disk integration is a means of calculating the Volume of a Solid of revolution, when integrating along the axis of revolution Shell integration (the shell method in Integral calculus) is a means of calculating the Volume of a Solid of revolution, when integrating In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires In Mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions In Integral calculus, the use of Partial fractions is required to integrate the general Rational function. In Calculus, interchange of the order of integration is a methodology that transforms multiple integrations of functions into other hopefully simpler integrals by For example, the Riemann integral can easily integrate density to find the mass of a steel beam, but cannot accommodate a steel ball resting on it. This motivates other definitions, under which a broader assortment of functions is integrable (Rudin 1987). The Lebesgue integral, in particular, achieves great flexibility by directing attention to the weights in the weighted sum.

The definition of the Lebesgue integral thus begins with a measure, μ. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with 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 Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.

To exploit this flexibility, Lebesgue integrals reverse the approach to the weighted sum. 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".

One common approach first defines the integral of the indicator function of a measurable set A by:

$\int 1_A d\mu = \mu(A)$ . In Mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with

This extends by linearity to a measurable simple function s, which attains only a finite number, n, of distinct non-negative values:

$\begin{align} \int s \, d\mu &{}= \int\left(\sum_{i=1}^{n} a_i 1_{A_i}\right) d\mu \\ &{}= \sum_{i=1}^{n} a_i\int 1_{A_i} \, d\mu \\ &{}= \sum_{i=1}^{n} a_i \, \mu(A_i) \end{align}$

(where the image of A_i under the simple function s is the constant value a_i). In mathematical field of Real analysis, a simple function is a real -valued function over a subset of the Real line which attains only a finite Thus if E is a measurable set one defines

$\int_E s \, d\mu = \sum_{i=1}^{n} a_i \, \mu(A_i \cap E) .$

Then for any non-negative measurable function f one defines

$\int_E f \, d\mu = \sup\left\{\int_E s \, d\mu\, \colon 0 \leq s\leq f\text{ and } s\text{ is a simple function}\right\};$

that is, the integral of f is set to be the supremum of all the integrals of simple functions that are less than or equal to f. In Mathematics, measurable functions are Well-behaved functions between measurable spaces. A general measurable function f, is split into its positive and negative values by defining

$\begin{align} f^+(x) &{}= \begin{cases} f(x), & \text{if } f(x) > 0 \\ 0, & \text{otherwise} \end{cases} \\ f^-(x) &{}= \begin{cases} -f(x), & \text{if } f(x) < 0 \\ 0, & \text{otherwise} \end{cases} \end{align}$

Finally, f is Lebesgue integrable if

$\int_E |f| \, d\mu < \infty , \,\!$

and then the integral is defined by

$\int_E f \, d\mu = \int_E f^+ \, d\mu - \int_E f^- \, d\mu . \,\!$

When the measure space on which the functions are defined is also a locally compact topological space (as is the case with the real numbers R), measures compatible with the topology in a suitable sense (Radon measures, of which the Lebesgue measure is an example) and integral with respect to them can be defined differently, starting from the integrals of continuous functions with compact support. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a Radon measure, named after Johann Radon, on a Hausdorff topological space X is defined in Measure theory to be a In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set More precisely, the compactly supported functions form a vector space that carries a natural topology, and a (Radon) measure can be defined as any continuous linear functional on this space; the value of a measure at a compactly supported function is then also by definition the integral of the function. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that One then proceeds to expand the measure (the integral) to more general functions by continuity, and defines the measure of a set as the integral of its indicator function. This is the approach taken by Bourbaki (2004) and a certain number of other authors. For details see Radon measures. In Mathematics, a Radon measure, named after Johann Radon, on a Hausdorff topological space X is defined in Measure theory to be a

Other integrals

Although the Riemann and Lebesgue integrals are the most important definitions of the integral, a number of others exist, including:

The Riemann-Stieltjes integral, an extension of the Riemann integral. In Mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes
The Lebesgue-Stieltjes integral, further developed by Johann Radon, which generalizes the Riemann-Stieltjes and Lebesgue integrals. In measure-theoretic analysis and related branches of Mathematics, Lebesgue-Stieltjes integration generalizes Riemann-Stieltjes and Lebesgue Johann Karl August Radon ( December 16, 1887 &ndash May 25, 1956) was an Austrian Mathematician. In Mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of
The Daniell integral, which subsumes the Lebesgue integral and Lebesgue-Stieltjes integral without the dependence on measures. One of the main difficulties with the traditional formulation of the Lebesgue integral is that it requires the initial development of a workable measure theory before any useful results In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of In measure-theoretic analysis and related branches of Mathematics, Lebesgue-Stieltjes integration generalizes Riemann-Stieltjes and Lebesgue In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with
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. Arnaud Denjoy ( 5 January 1884 &ndash 21 January 1974) was a French Mathematician. "Perron" redirects here For the cardinal see Jacques-Davy Duperron. Jaroslav Kurzweil (born 1926 (ˈjarɔslaf ˈkurt͡svajl is a Czech Mathematician. Ralph Henstock ( June 2 1923 &ndash January 17 2007) was an English Mathematician and author
The Itō integral and Stratonovich integral, which define integration with respect to stochastic processes such as Brownian motion. Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to Stochastic processes such as Brownian motion ( Wiener process) In Stochastic processes the Stratonovich integral (developed simultaneously by Ruslan L In Mathematics, the Wiener process is a continuous-time Stochastic process named in honor of Norbert Wiener.

Properties of integration

Linearity

The collection of Riemann integrable 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 integration

$f \mapsto \int_a^b f \; dx$

is a linear functional on this vector space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional Thus, firstly, the collection of integrable functions is closed under taking linear combinations; and, secondly, the integral of a linear combination is the linear combination of the integrals,

$\int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \,$

Similarly, the set of real-valued Lebesgue integrable functions on a given measure space E with measure μ is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral

$f\mapsto \int_E f d\mu$

is a linear functional on this vector space, so that

$\int_E (\alpha f + \beta g) \, d\mu = \alpha \int_E f \, d\mu + \beta \int_E g \, d\mu.$

More generally, consider the vector space of all measurable functions on a measure space (E,μ), taking values in a locally compact complete topological vector space V over a locally compact topological field K, f : E → V. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics In Mathematics, the real numbers may be described informally in several different ways In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematics, measurable functions are Well-behaved functions between measurable spaces. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. In Mathematics, a topological ring is a ring R which is also a Topological space such that both the addition and the multiplication are Then one may define an abstract integration map assigning to each function f an element of V or the symbol ∞,

$f\mapsto\int_E f d\mu, \,$

that is compatible with linear combinations. In this situation the linearity holds for the subspace of functions whose integral is an element of V (i. e. "finite"). The most important special cases arise when K is R, C, or a finite extension of the field Q_p of p-adic numbers, and V is a finite-dimensional vector space over K, and when K=C and V is a complex Hilbert space. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 This article assumes some familiarity with Analytic geometry and the concept of a limit.

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 real-valued functions on a set X, generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space. One of the main difficulties with the traditional formulation of the Lebesgue integral is that it requires the initial development of a workable measure theory before any useful results Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition See (Hildebrandt 1953) for an axiomatic characterisation of the integral.

Inequalities for integrals

A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval [a, b] and can be generalized to other notions of integral (Lebesgue and Daniell). The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Mathematical analysis and related areas of Mathematics, a set is called bounded, if it is in a certain sense of finite size In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set

Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. In Mathematics, a function f defined on some set X with real or complex values is called bounded, if the set Thus there are real numbers m and M so that m ≤ f (x) ≤ M for all x in [a, b]. In Mathematics, the real numbers may be described informally in several different ways 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

$m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a).$

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

$\int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx.$

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].

Subintervals. If [c, d] is a subinterval of [a, b] and f(x) is non-negative for all x, then

$\int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx.$

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:

$(fg)(x)= f(x) g(x), \; f^2 (x) = (f(x))^2, \; |f| (x) = |f(x)|.\,$

If f is Riemann-integrable on [a, b] then the same is true for |f|, and

$\left| \int_a^b f(x) \, dx \right| \leq \int_a^b | f(x) | \, dx.$

Moreover, if f and g are both Riemann-integrable then f ², g ², and fg are also Riemann-integrable, and

$\left( \int_a^b (fg)(x) \, dx \right)^2 \leq \left( \int_a^b f(x)^2 \, dx \right) \left( \int_a^b g(x)^2 \, dx \right).$

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 square-integrable functions f and g on the interval [a, b]. The pointwise product of two functions is another function obtained by multiplying the image of the two functions at each value in the domain In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In Mathematics, the Cauchy–Schwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the Cauchy–Schwarz–Bunyakovsky This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product.

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 Riemann-integrable functions. Then the functions |f|^p and |g|^q are also integrable and the following Hölder's inequality holds:

$\left|\int f(x)g(x)\,dx\right| \leq \left(\int \left|f(x)\right|^p\,dx \right)^{1/p} \left(\int\left|g(x)\right|^q\,dx\right)^{1/q}.$

For p = q = 2, Hölder's inequality becomes the Cauchy–Schwarz inequality. In Mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental Inequality between integrals and an indispensable tool

Minkowski inequality. Suppose that p ≥ 1 is a real number and f and g are Riemann-integrable functions. Then |f|^p, |g|^p and |f + g|^p are also Riemann integrable and the following Minkowski inequality holds:

$\left(\int \left|f(x)+g(x)\right|^p\,dx \right)^{1/p} \leq \left(\int \left|f(x)\right|^p\,dx \right)^{1/p} + \left(\int \left|g(x)\right|^p\,dx \right)^{1/p}.$

An analogue of this inequality for Lebesgue integral is used in construction of L^p spaces. In Mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are Normed vector spaces Let S be a Measure In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding

Conventions

In this section f is a real-valued Riemann-integrable function. In Mathematics, the real numbers may be described informally in several different ways The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The integral

$\int_a^b f(x) \, dx$

over an interval [a, b] is defined if a < b. This means that the upper and lower sums of the function f are evaluated on a partition a = x₀ ≤ x₁ ≤ . . . ≤ x_n = b whose values x_i are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating f within intervals [x_i , x_i +1] where an interval with a higher index lies to the right of one with a lower index. The values a and b, the end-points of the interval, are called the limits of integration of f. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Calculus and Mathematical analysis the limits of integration of the Integral \int_a^b f(x \ dx of a Riemann integrable Integrals can also be defined if a > b:

Reversing limits of integration. If a > b then define

$\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx.$

This, with a = b, implies:

Integrals over intervals of length zero. If a is a real number then

$\int_a^a f(x) \, dx = 0.$

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 a point, should be zero. In Mathematics, the real numbers may be described informally in several different ways In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume One reason for the first convention is that the integrability of f on an interval [a, b] implies that f is integrable on any subinterval [c, d], but in particular integrals have the property that:

Additivity of integration on intervals. If c is any element of [a, b], then

$\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.$

With the first convention the resulting relation

$\begin{align} \int_a^c f(x) \, dx &{}= \int_a^b f(x) \, dx - \int_c^b f(x) \, dx \\ &{} = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx \end{align}$

is then well-defined for any cyclic permutation of a, b, and c. In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the

Instead of viewing the above as conventions, one can also adopt the point of view that integration is performed on oriented manifolds only. A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back If M is such an oriented m-dimensional manifold, and M' is the same manifold with opposed orientation and ω is an m-form, then one has (see below for integration of differential forms):

$\int_M \omega = - \int_{M'} \omega \,.$

Fundamental theorem of calculus

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. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output 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. In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative

Statements of theorems

Fundamental theorem of calculus. Let f be a real-valued integrable function defined on a closed interval [a, b]. In Mathematics, the real numbers may be described informally in several different ways The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set If F is defined for x in [a, b] by

$F(x) = \int_a^x f(t)\, dt.$

then F is continuous on [a, b]. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output If f is continuous at x in [a, b], then F is differentiable at x, and F ′(x) = f(x). Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change

Second fundamental theorem of calculus. Let f be a real-valued integrable function defined on a closed interval [a, b]. If F is a function such that F ′(x) = f(x) for all x in [a, b] (that is, F is an antiderivative of f), then

$\int_a^b f(t)\, dt = F(b) - F(a).$

Corollary. In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative If f is a continuous function on [a, b], then f is integrable on [a, b], and F, defined by

$F(x) = \int_a^x f(t) \, dt$

is an anti-derivative of f on [a, b]. Moreover,

$\int_a^b f(t) \, dt = F(b) - F(a).$

Extensions

Improper integrals

Main article: Improper integral

The improper integral
$\int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} = \pi$
has unbounded intervals for both domain and range. In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified

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. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Mathematics, a sequence is an ordered list of objects (or events In the branch of Mathematics known as Real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition 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.

$\int_{a}^{\infty} f(x)dx = \lim_{b \to \infty} \int_{a}^{b} f(x)dx$

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.

$\int_{a}^{b} f(x)dx = \lim_{\epsilon \to 0} \int_{a+\epsilon}^{b} f(x)dx$

That is, the improper integral is the limit of proper integrals as one endpoint of the interval of integration approaches either a specified real number, or ∞, or −∞. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Mathematics, the real numbers may be described informally in several different ways In more complicated cases, limits are required at both endpoints, or at interior points.

Consider, for example, the function $\tfrac{1}{(x+1)\sqrt{x}}$ integrated from 0 to ∞ (shown right). At the lower bound, as x goes to 0 the function goes to ∞, and the upper bound is itself ∞, though the function goes to 0. Thus this is a doubly improper integral. Integrated, say, from 1 to 3, an ordinary Riemann sum suffices to produce a result of $\tfrac{\pi}{6}$ . To integrate from 1 to ∞, a Riemann sum is not possible. However, any finite upper bound, say t (with t > 1), gives a well-defined result, $\tfrac{\pi}{2} - 2\arctan \tfrac{1}{\sqrt{t}}$ . This has a finite limit as t goes to infinity, namely $\tfrac{\pi}{2}$ . Similarly, the integral from ¹⁄₃ to 1 allows a Riemann sum as well, coincidentally again producing $\tfrac{\pi}{6}$ . Replacing ¹⁄₃ by an arbitrary positive value s (with s < 1) is equally safe, giving $-\tfrac{\pi}{2} + 2\arctan\tfrac{1}{\sqrt{s}}$ . This, too, has a finite limit as s goes to zero, namely $\tfrac{\pi}{2}$ . Combining the limits of the two fragments, the result of this improper integral is

$\begin{align} \int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} &{} = \lim_{s \to 0} \int_{s}^{1} \frac{dx}{(x+1)\sqrt{x}} + \lim_{t \to \infty} \int_{1}^{t} \frac{dx}{(x+1)\sqrt{x}} \\ &{} = \lim_{s \to 0} \left( - \frac{\pi}{2} + 2 \arctan\frac{1}{\sqrt{s}} \right) + \lim_{t \to \infty} \left( \frac{\pi}{2} - 2 \arctan\frac{1}{\sqrt{t}} \right) \\ &{} = \frac{\pi}{2} + \frac{\pi}{2} \\ &{} = \pi . \end{align}$

This process is not guaranteed success; a limit may fail to exist, or may be unbounded. For example, over the bounded interval 0 to 1 the integral of $\tfrac{1}{x^2}$ does not converge; and over the unbounded interval 1 to ∞ the integral of $\tfrac{1}{\sqrt{x}}$ does not converge.

It may also happen that an integrand is unbounded at an interior point, in which case the integral must be split at that point, and the limit integrals on both sides must exist and must be bounded. Thus

$\begin{align} \int_{-1}^{1} \frac{dx}{\sqrt[3]{x^2}} &{} = \lim_{s \to 0} \int_{-1}^{-s} \frac{dx}{\sqrt[3]{x^2}} + \lim_{t \to 0} \int_{t}^{1} \frac{dx}{\sqrt[3]{x^2}} \\ &{} = \lim_{s \to 0} 3(1-\sqrt[3]{s}) + \lim_{t \to 0} 3(1-\sqrt[3]{t}) \\ &{} = 3 + 3 \\ &{} = 6. \end{align}$

But the similar integral

$\int_{-1}^{1} \frac{dx}{x} \,\!$

cannot be assigned a value in this way, as the integrals above and below zero do not independently converge. (However, see Cauchy principal value. In Mathematics, the Cauchy principal value of certain Improper integrals named after Augustin Louis Cauchy, is defined as either the finite )

Multiple integration

Main article: Multiple integral

Double integral as volume under a surface. The multiple integral is a type of definite Integral extended to functions of more than one real Variable, for example

Integrals can be taken over regions other than intervals. In general, an integral over a set E of a function f is written:

$\int_E f(x) \, dx.$

Here x need not be a real number, but can be another suitable quantity, for instance, a vector in R³. Fubini's theorem shows that such integrals can be rewritten as an iterated integral. In Mathematical analysis, Fubini's theorem, named after Guido Fubini, states that if \int_{A\times B} |f(xy|\d(xy The multiple integral is a type of definite Integral extended to functions of more than one real Variable, for example In other words, the integral can be calculated by integrating one coordinate at a time.

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 x-axis, 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. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined (The same volume can be obtained via the triple integral — the integral of a function in three variables — of the constant function f(x, y, z) = 1 over the above-mentioned region between the surface and the plane. ) If the number of variables is higher, then the integral represents a hypervolume, a volume of a solid of more than three dimensions that cannot be graphed. In Physics and Mathematics, a sequence of n numbers can be understood as a location in an n -dimensional space

For example, the volume of the parallelepiped of sides 4 × 6 × 5 may be obtained in two ways:

By the double integral

$\iint_D 5 \ dx\, dy$

of the function f(x, y) = 5 calculated in the region D in the xy-plane which is the base of the parallelepiped. Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism

By the triple integral

$\iiint_\mathrm{parallelepiped} 1 \, dx\, dy\, dz$

of the constant function 1 calculated on the parallelepiped itself.

Because it is impossible to calculate the antiderivative of a function of more than one variable, indefinite multiple integrals do not exist, so such integrals are all definite. In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative

Line integrals

Main article: Line integral

A line integral sums together elements along a curve. In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated

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. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space.

A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object 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. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. 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). Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In differential calculus, a differential is traditionally an Infinitesimally small change in a Variable. This weighting distinguishes the line integral from simpler integrals defined on intervals. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that work is equal to force multiplied by distance may be expressed (in terms of vector quantities) as:

$W=\vec F\cdot\vec d$ ;

which is paralleled by the line integral:

$W=\int_C \vec F\cdot d\vec s$ ;

which sums up vector components along a continuous path, and thus finds the work done on an object moving through a field, such as an electric or gravitational field

Surface integrals

Main article: Surface integral

The definition of surface integral relies on splitting the surface into small surface elements. In Physics, mechanical work is the amount of Energy transferred by a Force. In Physics, a force is whatever can cause an object with Mass to Accelerate. In Mathematics, a surface integral is a Definite integral taken over a Surface (which may be a curved set in Space) it can be thought

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. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another The multiple integral is a type of definite Integral extended to functions of more than one real Variable, for example In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated The function to be integrated may be a scalar field or a vector field. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. 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. In the various subfields of Physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks 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:

$\int_S {\mathbf v}\cdot \,d{\mathbf {S}}$ . In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R

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 classical theory of electromagnetism. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Classical theory has at least two distinct meanings in Physics: In the context of Quantum mechanics, "classical theory" refers to Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of

Integrals of differential forms

Main article: differential form

A differential form is a mathematical concept in the fields of multivariable calculus, differential topology and tensors. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is Multivariable calculus is the extension of Calculus in one Variable to calculus in several variables the functions which are differentiated and integrated involve In Mathematics, differential topology is the field dealing with differentiable functions on Differentiable manifolds It is closely related to Differential History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually The modern notation for the differential form, as well as the idea of the differential forms as being the wedge products of exterior derivatives forming an exterior algebra, was introduced by Élie Cartan. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms Élie Joseph Cartan ( 9 April 1869 &ndash 6 May 1951) was an influential French Mathematician, who did fundamental

We initially work in an open set in Rⁿ. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in A 0-form is defined to be a smooth function f. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability When we integrate a function f over an m-dimensional subspace S of Rⁿ, we write it as

$\int_S f\,dx^1 \cdots dx^m.$

(The superscripts are indices, not exponents. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it ) We can consider dx¹ through dxⁿ to be formal objects themselves, rather than tags appended to make integrals look like Riemann sums. In Mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph otherwise known as an Integral. Alternatively, we can view them as covectors, and thus a measure of "density" (hence integrable in a general sense). In Linear algebra, a one-form on a Vector space is the same as a Linear functional on the space In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with We call the dx¹, …,dxⁿ basic 1-forms. In Linear algebra, a one-form on a Vector space is the same as a Linear functional on the space

We define the wedge product, "∧", a bilinear "multiplication" operator on these elements, with the alternating property that

$dx^a \wedge dx^a = 0 \,\!$

for all indices a. Note that alternation along with linearity implies dx^b∧dx^a = −dx^a∧dx^b. This also ensures that the result of the wedge product has an orientation. See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which

We define the set of all these products to be basic 2-forms, and similarly we define the set of products of the form dx^a∧dx^b∧dx^c to be basic 3-forms. A general k-form is then a weighted sum of basic k-forms, where the weights are the smooth functions f. Together these form a vector space with basic k-forms as the basis vectors, and 0-forms (smooth functions) as the field of scalars. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The wedge product then extends to k-forms in the natural way. Over Rⁿ at most n covectors can be linearly independent, thus a k-form with k > n will always be zero, by the alternating property.

In addition to the wedge product, there is also the exterior derivative operator d. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms This operator maps k-forms to (k+1)-forms. For a k-form ω = f dx^a over Rⁿ, we define the action of d by:

${\bold d}{\omega} = \sum_{i=1}^n \frac{\partial f}{\partial x_i} dx^i \wedge dx^a.$

with extension to general k-forms occurring linearly.

This more general approach allows for a more natural coordinate-free approach to integration on manifolds. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be It also allows for a natural generalisation of the fundamental theorem of calculus, called Stokes' theorem, which we may state as

$\int_{\Omega} {\bold d}\omega = \int_{\partial\Omega} \omega \,\!$

where ω is a general k-form, and ∂Ω denotes the boundary of the region Ω. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from For a different notion of boundary related to Manifolds see that article Thus in the case that ω is a 0-form and Ω is a closed interval of the real line, this reduces to the fundamental theorem of calculus. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In the case that ω is a 1-form and Ω is a 2-dimensional region in the plane, the theorem reduces to Green's theorem. In Physics and Mathematics, Green's theorem gives the relationship between a Line integral around a simple closed curve C and a Double integral Similarly, using 2-forms, and 3-forms and Hodge duality, we can arrive at Stokes' theorem and the divergence theorem. In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail In this way we can see that differential forms provide a powerful unifying view of integration.

Methods and applications

Computing integrals

The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. It proceeds like this:

Let f(x) be the function of x to be integrated over a given interval [a, b].
Find an antiderivative of f, that is, a function F such that F' = f on the interval.
Then, by the fundamental theorem of calculus, provided the integrand and integral have no singularities on the path of integration,

$\int_a^b f(x)\,dx = F(b)-F(a).$

Note that the integral is not actually the antiderivative, but the fundamental theorem allows us to use antiderivatives to evaluate definite integrals. In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be

The difficult step is often finding an 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:

Even if these techniques fail, it may still be possible to evaluate a given integral. In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other In Calculus, interchange of the order of integration is a methodology that transforms multiple integrations of functions into other hopefully simpler integrals by In Mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions In Integral calculus, the use of Partial fractions is required to integrate the general Rational function. Integration by reduction formulae can be used when we want to integrate a function raised to the power n Differentiation under the integral sign is a useful operation in the Mathematical field of Calculus. The next most common technique is residue calculus, whilst for nonelementary integrals Taylor series can sometimes be used to find the antiderivative. In Complex analysis, the residue is a Complex number which describes the behavior of Line integrals of a Meromorphic function around a singularity In Mathematics, a nonelementary integral is an Integral for which it can be shown that there exists no formula in terms of Elementary functions ' (i In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives The method of convolution using Meijer G-functions can also be used, assuming that the integrand can be written as a product of Meijer G-functions. The G-function was defined for the first time by the Dutch mathematician Cornelis Simon Meijer (1904-1974 in 1936 as an attempt to introduce a very general function that includes 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. In Mathematical analysis, Parseval's identity is a fundamental result on the Summability of the Fourier series of a function Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral. The Gaussian integral, or probability integral, is the Improper integral of the Gaussian function e^ over the entire real line

Computations of volumes of solids of revolution can usually be done with disk integration or shell integration. In Mathematics, Engineering, and Manufacturing, a solid of revolution is a solid figure obtained by rotating a Plane curve around Disk integration is a means of calculating the Volume of a Solid of revolution, when integrating along the axis of revolution Shell integration (the shell method in Integral calculus) is a means of calculating the Volume of a Solid of revolution, when integrating

Specific results which have been worked out by various techniques are collected in the list of integrals. See the following pages for lists of Integrals: List of integrals of rational functions List of integrals of irrational functions

Symbolic algorithms

Main article: Symbolic integration

Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Symbolic integration is the problem of finding a formula for the Antiderivative, or indefinite integral, of a given function f ( x) i Extensive tables of integrals have been compiled and published over the years for this purpose. See the following pages for lists of Integrals: List of integrals of rational functions List of integrals of irrational functions 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. A computer is a Machine that manipulates data according to a list of instructions. A computer algebra system ( CAS) is a software program that facilitates Symbolic mathematics. Symbolic integration presents a special challenge in the development of such systems.

A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather simple-looking function does not exist. For instance, it is known that the antiderivatives of the functions exp ( x²), 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. In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce This article discusses the concept of elementary functions in differential algebra Differential Galois theory provides general criteria that allow one to determine whether the antiderivative of an elementary function is elementary. In Mathematics, the Antiderivatives of certain Elementary functions cannot themselves be expressed as elementary functions 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. The Risch algorithm, named after Robert H Risch is an Algorithm for the Calculus operation of indefinite integration (i Mathematica is a computer program used widely in scientific engineering and mathematical fields A computer algebra system ( CAS) is a software program that facilitates Symbolic mathematics. In Mathematics, an n th root of a Number a is a number b such that bn = a.

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 and so on). Special functions are particular mathematical functions which have more or less established names and notations due to their importance for the Mathematical analysis Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Note This article describes a very general class of functions In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function Extending the Risch-Norman algorithm so that it includes these functions is possible but challenging.

Most humans are not able to integrate such general formulae, so in a sense computers are more skilled at integrating highly complicated formulae. Very complex formulae are unlikely to have closed-form antiderivatives, so how much of an advantage this presents is a philosophical question that is open for debate.

Numerical quadrature

Main article: numerical integration

The integrals encountered in a basic calculus course are deliberately chosen for simplicity; those found in real applications are not always so accommodating. In Numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite Integral, and by extension Some integrals cannot be found exactly, some 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 methods for approximating integrals, which today use floating point arithmetic on digital electronic computers. In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. A computer is a Machine that manipulates data according to a list of instructions. Many of the ideas arose much earlier, for hand calculations; but the speed of general-purpose computers like the ENIAC created a need for improvements. ENIAC, short for Electronic Numerical Integrator And Computer, was the first general-purpose electronic Computer.

The goals of numerical integration are accuracy, reliability, efficiency, and generality. Sophisticated methods can vastly outperform a naive method by all four measures (Dahlquist & Björck forthcoming; Kahaner, Moler & Nash 1989; Stoer & Bulirsch 2002). Consider, for example, the integral

$\int_{-2}^{2} \tfrac15 \left( \tfrac{1}{100}(322 + 3 x (98 + x (37 + x))) - 24 \frac{x}{1+x^2} \right) dx ,$

which has the exact answer ⁹⁴⁄₂₅ = 3. 76. (In ordinary practice the answer is not known in advance, so an important task — not explored here — is to decide when an approximation is good enough. ) A “calculus book” approach divides the integration range into, say, 16 equal pieces, and computes function values.

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

Numerical quadrature methods: ■ Rectangle, ■ Trapezoid, ■ Romberg, ■ Gauss

Using the left end of each piece, the rectangle method sums 16 function values and multiplies by the step width, h, here 0. In Mathematics, specifically in Integral calculus, the rectangle method (also called the Midpoint or Mid-Ordinate Rule) uses an Approximation 25, to get an approximate value of 3. 94325 for the integral. The accuracy is not impressive, but calculus formally uses pieces of infinitesimal width, so initially this may seem little cause for concern. Indeed, repeatedly doubling the number of steps eventually produces an approximation of 3. 76001. However 2¹⁸ pieces are required, a great computational expense for so little accuracy; and a reach for greater accuracy can force steps so small that arithmetic precision becomes an obstacle.

A better approach replaces the horizontal tops of the rectangles with slanted tops touching the function at the ends of each piece. This trapezium 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. In Mathematics, the trapezium rule (the British term or trapezoidal rule (the American term is a way to approximately calculate the definite integral This immediately improves the approximation to 3. 76925, which is noticeably more accurate. Furthermore, only 2¹⁰ pieces are needed to achieve 3. 76000, substantially less computation than the rectangle method for comparable accuracy.

Romberg's method builds on the trapezoid method to great effect. In Numerical analysis, Romberg's method generates a triangular array consisting of numerical estimates of the definite integral \int_a^b f(x First, the step lengths are halved incrementally, giving trapezoid approximations denoted by T(h₀), T(h₁), 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). In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of 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. In Numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation Polynomial for a given set of data points 00,6. 128), (2. 00,4. 352), (1. 00,3. 908)} is 3. 76+0. 148h², producing the extrapolated value 3. 76 at h = 0.

Gaussian quadrature often requires noticeably less work for superior accuracy. In Numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a Weighted sum of function 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 error analysis, and a little luck. An n-point 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. )

Shifting the range left a little, so the integral is from −2. 25 to 1. 75, removes the symmetry. Nevertheless, the trapezoid method is rather slow, the polynomial interpolation method of Romberg is acceptable, and the Gaussian method requires the least work — if the number of points is known in advance. As well, rational interpolation can use the same trapezoid evaluations as the Romberg method to greater effect.

Quadrature method cost comparison
Method	Trapezoid	Romberg	Rational	Gauss
Points	1048577	257	129	36
Rel. Err.	−5. 3×10⁻¹³	−6. 3×10⁻¹⁵	8. 8×10⁻¹⁵	3. 1×10⁻¹⁵
Value	$\textstyle \int_{-2.25}^{1.75} f(x)\,dx = 4.1639019006585897075\ldots$

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 hybrid. Symmetry can still be exploited by splitting this integral into two ranges, from −2. 25 to −1. 75 (no symmetry), and from −1. 75 to 1. 75 (symmetry). More broadly, adaptive quadrature partitions a range into pieces based on function properties, so that data points are concentrated where they are needed most. In Applied mathematics, adaptive quadrature is a process in which the Integral of a function f(x is approximated using static

This brief introduction omits higher-dimensional integrals (for example, area and volume calculations), where alternatives such as Monte Carlo integration have great importance. In Mathematics, Monte Carlo integration is Numerical quadrature using pseudorandom numbers That is Monte Carlo integration methods are Algorithms

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.

References

Apostol, Tom M. (1967), Calculus, Vol. The multiple integral is a type of definite Integral extended to functions of more than one real Variable, for example In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative In Numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite Integral, and by extension In Mathematics, an integral equation is an equation in which an unknown function appears under an Integral sign In the branch of Mathematics known as Real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the Integral In Mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph otherwise known as an Integral. Product integrals are a multiplicative version of standard Integrals of infinitesimal calculus Tom Mike Apostol (born 1923 is an American analytic number theorist and professor at the California Institute of Technology. 1: One-Variable Calculus with an Introduction to Linear Algebra (2nd ed. ), Wiley, ISBN 978-0-471-00005-1
Bourbaki, Nicolas (2004), Integration I, Springer Verlag, ISBN 3-540-41129-1 . John Wiley & Sons Inc, also referred to as Wiley, is a global Publishing company that markets its products to professionals and consumers students and instructors Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic In particular chapters III and IV.
Burton, David M. (2005), The History of Mathematics: An Introduction (6^th ed. ), McGraw-Hill, p. The McGraw-Hill Companies Inc, ( is a Publicly traded corporation headquartered in Rockefeller Center in New York City. p. 359, ISBN 978-0-07-305189-5
Cajori, Florian (1929), A History Of Mathematical Notations Volume II, Open Court Publishing, pp. Florian Cajori ( February 28 1859 in St Aignan (near Thusis Graubünden, Switzerland &mdash August 15, 1930, Berkeley The Open Court Publishing Company is a Publisher with offices in Chicago and La Salle Illinois. 247–252, ISBN 978-0-486-67766-8, <http://www.archive.org/details/historyofmathema027671mbp>
Dahlquist, Germund & Björck, Åke (forthcoming), “Chapter 5: Numerical Integration”, Numerical Methods in Scientific Computing, Philadelphia: SIAM, <http://www.mai.liu.se/~akbjo/NMbook.html>
Folland, Gerald B. Germund Dahlquist (January 16 1925 Uppsala - February 8 2005 Stockholm) was a Swedish Mathematician known primarily for his early contributions (1984), Real Analysis: Modern Techniques and Their Applications (1st ed. ), John Wiley & Sons, ISBN 978-0-471-80958-6
Fourier, Jean Baptiste Joseph (1822), Théorie analytique de la chaleur, Chez Firmin Didot, père et fils, p. John Wiley & Sons Inc, also referred to as Wiley, is a global Publishing company that markets its products to professionals and consumers students and instructors Jean Baptiste Joseph Fourier ( March 21, 1768 &ndash May 16, 1830) was a French Mathematician and Physicist §231, <http://books.google.com/books?id=TDQJAAAAIAAJ>
Available in translation as Fourier, Joseph (1878), The analytical theory of heat, Cambridge University Press, pp. Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 pp. 200–201, <http://www.archive.org/details/analyticaltheory00fourrich>
Heath, T. L., ed. Sir Thomas Little Heath ( October 5, 1861 &ndash March 16, 1940) was a British civil servant Mathematician, classical (2002), The Works of Archimedes, Dover, ISBN 978-0-486-42084-4, <http://www.archive.org/details/worksofarchimede029517mbp>
(Originally published by Cambridge University Press, 1897, based on J. Dover Publications is an American book Publisher founded in 1941 by Hayward Cirker and his wife Blanche Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 L. Heiberg's Greek version. )
Hildebrandt, T. H. (1953), “Integration in abstract spaces”, Bulletin of the American Mathematical Society 59 (2): 111–139, ISSN 0273-0979, <http://projecteuclid.org/euclid.bams/1183517761>
Kahaner, David; Moler, Cleve & Nash, Stephen (1989), “Chapter 5: Numerical Quadrature”, Numerical Methods and Software, Prentice Hall, ISBN 978-0-13-627258-8
Leibniz, Gottfried Wilhelm (1899), Gerhardt, Karl Immanuel, ed. Bulletin of the American Mathematical Society (often abbreviated as Bull An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. Cleve Barry Moler is a mathematician and computer programmer specializing in Numerical analysis. Prentice Hall is a leading educational publisher It is an Imprint of Pearson Education Inc , Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern. Erster Band, Berlin: Mayer & Müller, <http://name.umdl.umich.edu/AAX2762.0001.001>
Miller, Jeff, Earliest Uses of Symbols of Calculus, <http://members.aol.com/jeff570/calculus.html>. Retrieved on 2 June 2007
O’Connor, J. J. & Robertson, E. F. (1996), A history of the calculus, <http://www-history.mcs.st-andrews.ac.uk/HistTopics/The_rise_of_calculus.html>. Retrieved on 9 July 2007
Rudin, Walter (1987), “Chapter 1: Abstract Integration”, Real and Complex Analysis (International ed. Walter Rudin (born 1921) is an American Mathematician, currently a Professor Emeritus of Mathematics at the University of Wisconsin-Madison ), McGraw-Hill, ISBN 978-0-07-100276-9
Saks, Stanisław (1964), Theory of the integral (English translation by L. The McGraw-Hill Companies Inc, ( is a Publicly traded corporation headquartered in Rockefeller Center in New York City. Stanisław Saks (1897 — 1942 was a Polish Mathematician and university tutor known primarily for his membership in the Scottish Café circle an extensive C. Young. With two additional notes by Stefan Banach. Second revised ed. ), New York: Dover, <http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez=>
Stoer, Josef & Bulirsch, Roland (2002), “Chapter 3: Topics in Integration”, Introduction to Numerical Analysis (3rd ed. ), Springer, ISBN 978-0-387-95452-3 . Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic
W3C (2006), Arabic mathematical notation, <http://www.w3.org/TR/arabic-math/>

External links

The Integrator by Wolfram Research
Function Calculator from WIMS
Mathematical Assistant on Web online calculation of integrals, allows to integrate in small steps (includes also hints for next step which cover techniques like by parts, substitution, partial fractions, application of formulas and others, powered by Maxima_(software))
P. For other meanings of Maxima see Maxima Maxima is a free Computer algebra system based on a 1982 version of S. Wang, Evaluation of Definite Integrals by Symbolic Manipulation (1972) - a cookbook of definite integral techniques
Definite Integrals
Online Integral Calculator

Online books

Keisler, H. Jerome, Elementary Calculus: An Approach Using Infinitesimals, University of Wisconsin
Stroyan, K. D. , A Brief Introduction to Infinitesimal Calculus, University of Iowa
Mauch, Sean, Sean's Applied Math Book, CIT, an online textbook that includes a complete introduction to calculus
Crowell, Benjamin, Calculus, Fullerton College, an online textbook
Garrett, Paul, Notes on First-Year Calculus
Hussain, Faraz, Understanding Calculus, an online textbook
Kowalk, W. P. , Integration Theory, University of Oldenburg. A new concept to an old problem. Online textbook
Sloughter, Dan, Difference Equations to Differential Equations, an introduction to calculus
Wikibook of Calculus
Numerical Methods of Integration at Holistic Numerical Methods Institute

Dictionary

-adjective

Constituting a whole together with other parts or factors; not omittable or removable
(mathematics) Of or relating to an integer.

-noun

(mathematics) A numerical measure computed by a limiting process in which the domain of a function is divided into small subintervals and the value of the function at a point in each subinterval is multiplied by the measurement of that subinterval, all these products then being summed.
(mathematics) the result of summation of the product of a function and an infinitesimal.

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