Definite integral of a function represents 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. Given a function f(x) of a real variable x and an interval [a,b] of the real line, the integral For other uses, see Calculus (disambiguation). ... Look up integration in Wiktionary, the free dictionary. ... Look up integral in Wiktionary, the free dictionary. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1400 × 1400 pixel, file size: 58 KB, MIME type: image/png) This shows positive and negative areas of At time of uploading the SVG is rendered horribly by MediaWiki, but validates, and looks fine with ASV6 and... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1400 × 1400 pixel, file size: 58 KB, MIME type: image/png) This shows positive and negative areas of At time of uploading the SVG is rendered horribly by MediaWiki, but validates, and looks fine with ASV6 and... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other uses, see Calculus (disambiguation). ... Analysis has its beginnings in the rigorous formulation of calculus. ... This article is about functions in mathematics. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In computer science and mathematics, a variable (pronounced ) (sometimes called an object or identifier in computer science) is a symbolic representation used to denote a quantity or expression. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, the real line is simply the set of real numbers. ...

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. Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. ...

The term "integral" may also refer to the notion of antiderivative, a function F whose derivative is the given function f. 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. In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ... This article is about derivatives and differentiation in mathematical calculus. ...

The principles of integration were formulated by Isaac Newton and Gottfried Leibniz in the late seventeenth century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation, and the definite integral of a function can be easily computed once an antiderivative is known. Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Leibniz redirects here. ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... For other uses, see Calculus (disambiguation). ...

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. 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 a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. Integrals of differential forms play a fundamental role in modern differential 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. Modern concepts of integration are based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue. Bernhard Riemann. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ... An open surface with X-, Y-, and Z-contours shown. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Classical electrodynamics (or classical electromagnetism) is a theory of electromagnetism that was developed over the course of the 19th century, most prominently by James Clerk Maxwell. ... The integral of a positive function can be interpreted as the area under a curve. ... Henri Lebesgue Henri Léon Lebesgue (June 28, 1875, Beauvais – July 26, 1941, Paris) was a French mathematician, most famous for his theory of integration. ...

History

See also: History of calculus

// The method of integration can be traced back to the Egyptians, in the Moscow Mathematical Papyrus circa 1800 BC, which gives the formula for finding the volume of a pyramidal frustrum. ...

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. 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. 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. 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. This method was later used by Zu Chongzhi to find the volume of a sphere. Some ideas of integral calculus are found in the Siddhanta Shiromani, a 12th century astronomy text by Indian mathematician Bhāskara II. The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts discovered. ... For other meanings, see pyramid (disambiguation). ... A frustum is the portion of a solid â€“ normally a cone or pyramid â€“ which lies between two parallel planes cutting the solid. ... 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 area of the containing shape. ... Not to be confused with Eudoxus of Cyzicus. ... For other uses, see Archimedes (disambiguation). ... A possible likeness of Liu Hui on japanpostage stamp This is a Chinese name; the family name is 劉 (Liu) Liu Hui 劉徽 was a Chinese mathematician who lived in the 200s in the Wei Kingdom. ... Zu Chongzhi (Traditional Chinese: ; Simplified Chinese: ; Hanyu Pinyin: ZÇ” ChōngzhÄ«; Wade-Giles: Tsu Chung-chih, 429–500), courtesy name Wenyuan (文遠), was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties. ... Bhaskara (1114 – 1185), also known as Bhaskara II and Bhaskara Achārya (Bhaskara the teacher), was an Indian mathematician and astronomer. ...

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. 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. Coins illustrating Cavalieris principle Bonaventura Francesco Cavalieri (in Latin, Cavalerius) (1598–November 30, 1647) was an Italian mathematician best known today for Cavalieris principle, which states that the volumes of two objects are equal if the areas of corresponding cross-sections are in all cases equal. ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... Isaac Barrow (October 1630 - May 4, 1677) was an English divine, scholar and mathematician who is generally given minor credit for his role in the development of modern calculus; in particular, for his work regarding the tangent; for example, Barrow is given credit for being the first to calculate the... Evangelista Torricelli portrayed on the frontpage of Lezioni dEvangelista Torricelli. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ...

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 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. The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Leibniz redirects here. ... For other uses, see Calculus (disambiguation). ...

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". 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. 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. Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. For the second husband of Henrietta Howard, Countess of Suffolk, see George Berkeley (MP). ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Bernhard Riemann. ... Henri Lebesgue Henri Léon Lebesgue (June 28, 1875, Beauvais – July 26, 1941, Paris) was a French mathematician, most famous for his theory of integration. ... 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 physical sizes). ...

Notation

Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with or , which Newton used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted. Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ...

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 modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819–20, reprinted in his book of 1822 (Cajori 1929, pp. 249–250; Fourier 1822, §231). In Arabic mathematical notation which is written from right to left, an inverted integral symbol is used (W3C 2006). Leibniz redirects here. ... An italicized long s used in the word Congress in the United States Bill of Rights. ... Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... Modern Arabic mathematical notation is a mathematical notation that is based on the Arabic script. ... Image File history File links No higher resolution available. ...

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

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. More complicated cases may vary the notation slightly. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

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: Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

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

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

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 have been used to express the idea of objects so small that there is no way to see them or to measure them. ...

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. 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]. (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).) So the exact value of the area under the curve is computed formally as The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... This article is about derivatives and differentiation in mathematical calculus. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

The notation

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). Shortcomings of Riemann's dependence on intervals and continuity motivated newer definitions, especially the Lebesgue integral, which is founded on an ability to extend the idea of "measure" in much more flexible ways. Thus the notation Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... The integral of a positive function can be interpreted as the area under a curve. ...

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. 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, In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, a differential operator is an operator defined as a function of the differentiation operator. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

from which Green's theorem, the divergence theorem, and the fundamental theorem of calculus follow. In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special two-dimensional case of... In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the surface. ... 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. Not only do these methods vindicate the intuitions of the pioneers, they also lead to new mathematics. Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural...

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). 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. Let [a,b] be a closed interval of the real line; then a tagged partition of [a,b] is a finite sequence In the branch of mathematics known as real analysis, the Riemann integral ℛ, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1260 × 1260 pixel, file size: 38 KB, MIME type: image/png) An example Riemann sum for the integral showing tagged partition. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1260 × 1260 pixel, file size: 38 KB, MIME type: image/png) An example Riemann sum for the integral showing tagged partition. ... In mathematics, a Riemann sum is a method for approximating the values of integrals. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

Riemann sums converging as intervals halve, whether sampled at ■ right, ■ minimum, ■ maximum, or ■ left.

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 Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1260 × 1260 pixel, file size: 66 KB, MIME type: image/png) An example of Riemann sums for the integral sampling each interval at right (blue), minimum (red), maximum (green), or left (yellow). ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1260 × 1260 pixel, file size: 66 KB, MIME type: image/png) An example of Riemann sums for the integral sampling each interval at right (blue), minimum (red), maximum (green), or left (yellow). ...

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

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). 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 the simplest case, the Lebesgue measure μ(A) of an interval A = [a,b] is its width, b − a, so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals. 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 physical sizes). ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...

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: In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... 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 physical sizes). ...

.

This extends by linearity to a measurable simple function s, which attains only a finite number, n, of distinct non-negative values: In mathematics, especially in mathematical analysis, a simple function is a measurable function whose range is finite. ...

(where the image of A_i under the simple function s is the constant value a_i). Thus if E is a measurable set one defines

Then for any non-negative measurable function f one defines In mathematics, measurable functions are well-behaved functions between measurable spaces. ...

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. A general measurable function f, is split into its positive and negative values by defining In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...

Finally, f is Lebesgue integrable if

and then the integral is defined by

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. 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. 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 topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a Radon measure on a Hausdorff topological space X is a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular. ... 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 real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... 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 structures that allow one to formalize 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 preserves the operations of vector addition and scalar multiplication. ... In mathematics, a Radon measure on a Hausdorff topological space X is a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular. ...

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.
• The Lebesgue-Stieltjes integral, further developed by Johann Radon, which generalizes the Riemann-Stieltjes and Lebesgue integrals.
• The Daniell integral, which subsumes the Lebesgue integral and Lebesgue-Stieltjes integral without the dependence on measures.
• The Henstock-Kurzweil integral, variously defined by Arnaud Denjoy, Oskar Perron, and (most elegantly, as the gauge integral) Jaroslav Kurzweil, and developed by Ralph Henstock.
• The Itō integral and Stratonovich integral, which define integration with respect to stochastic processes such as Brownian motion.

In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. ... In measure-theoretic analysis and related branches of mathematics, Lebesgue-Stieltjes integration generalizes Riemann-Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework. ... Johann Radon (December 16, 1887–May 25, 1956) was a mathematician born in Litomerice in Bohemia (now Czech Republic). ... In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. ... The integral of a positive function can be interpreted as the area under a curve. ... The construction of the Lebesgue Integral is built on top of measure theory. ... The integral of a positive function can be interpreted as the area under a curve. ... In measure-theoretic analysis and related branches of mathematics, Lebesgue-Stieltjes integration generalizes Riemann-Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework. ... 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 physical sizes). ... In mathematics, the Henstock-Kurzweil integral, also known as the Denjoy integral (pronounce Denjua) and the Perron integral, is a possible definition of the integral of a function. ... Arnaud Denjoy (5 January 1884 – 21 January 1974) was a French mathematician. ... Oskar Perron (7 May 1880 – 22 February 1975) was a German mathematician. ... Jaroslav Kurzweil (1926 – ) is a Czech mathematician. ... Ralph Henstock (born June 2, 1923) is a British mathematician. ... Itō calculus, named after Kiyoshi Itō, treats mathematical operations on stochastic processes. ... The Stratonovich Integral is a stochastic integral, the commonest alternative to the Itō integral. ... A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process 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

is a linear functional on this vector space. 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,

• 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

is a linear functional on this vector space, so that

• 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. Then one may define an abstract integration map assigning to each function f an element of V or the symbol ∞,

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.

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. See (Hildebrandt 1953) for an axiomatic characterisation of the integral. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... 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 as numbers that can be given by an infinite decimal representation, such as 2. ... 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 physical sizes). ... 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 like a small portion of a compact space. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... 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 continuous as maps R × R → R, where R × R carries the product topology. ... In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... The construction of the Lebesgue Integral is built on top of measure theory. ... This article is about the group of mathematicians named Nicolas Bourbaki. ...

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). This article is about functions in mathematics. ... 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, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

• Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that m ≤ f (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(b − a) and M(b − a), it follows that

• Inequalities between functions. If f(x) ≤ g(x) for each x in [a, b] then each of the upper and lower sums of f is bounded above by the upper and lower sums, respectively, of g. Thus

This is a generalization of the above inequalities, as M(b − a) is the integral of the constant function with value M over [a, b].

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

• Products and absolute values of functions. If f and g are two functions then we may consider their pointwise products and powers, and absolute values:

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

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

This inequality, known as the Cauchy–Schwarz inequality, plays a prominent role in Hilbert space theory, where the left hand side is interpreted as the inner product of two square-integrable functions f and g on the interval [a, b].

• Hölder's inequality. Suppose that p and q are two real numbers, 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, and f and g are two Riemann-integrable functions. Then the functions |f|^p and |g|^q are also integrable and the following Hölder's inequality holds:

For p = q = 2, Hölder's inequality becomes the Cauchy–Schwarz inequality.

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

An analogue of this inequality for Lebesgue integral is used in construction of L^p spaces.

In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The pointwise product of two functions is another function, obtained by multiplying the results of the two functions. ... In mathematics, the absolute value (or modulus[1]) 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–Bunyakovski–Schwarz inequality, named after Augustin Louis Cauchy, Viktor Yakovlevich Bunyakovsky and Hermann Amandus Schwarz, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematical analysis, Hölders inequality, named after Otto Hölder, is a fundamental inequality relating Lp spaces: let S be a measure space, let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, let f be in Lp(S) and g be in Lq(S). ... In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

Conventions

In this section f is a real-valued Riemann-integrable function. The integral In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... This article is about functions in mathematics. ...

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. Integrals can also be defined if a > b: In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In calculus and mathematical analysis the limits of integration of the integral of a Riemann integrable function f defined on a closed and bounded interval [a, b] are the real numbers a and b. ...

• Reversing limits of integration. If a > b then define

This, with a = b, implies:

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

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. 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: In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... Zero redirects here. ...

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

With the first convention the resulting relation In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ...

is then well-defined for any cyclic permutation of a, b, and c.

Instead of viewing the above as conventions, one can also adopt the point of view that integration is performed on oriented manifolds only. 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): The torus is an orientable surface. ...

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. 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. The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... This article is about derivatives and differentiation in mathematical calculus. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

Statements of theorems

• Fundamental theorem of calculus. Let f be a real-valued integrable function defined on a closed interval [a, b]. If F is defined for x in [a, b] by

then F is continuous on [a, b]. If f is continuous at x in [a, b], then F is differentiable at x, and F ′(x) = f(x).

• 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

• Corollary. If f is a continuous function on [a, b], then f is integrable on [a, b], and F, defined by

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

In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... This article is about functions in mathematics. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

Extensions

Improper integrals

Main article: Improper integral

The improper integral

has unbounded intervals for both domain and range.

A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration. An improper integral occurs when one or more of these conditions is not satisfied. In some cases such integrals may be defined by considering the limit of a sequence of proper Riemann integrals on progressively larger intervals. It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... For other senses of this word, see sequence (disambiguation). ... In the branch of mathematics known as real analysis, the Riemann integral â„›, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. ...

If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity.

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.

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 more complicated cases, limits are required at both endpoints, or at interior points. Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Consider, for example, the function 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 . 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, . This has a finite limit as t goes to infinity, namely . Similarly, the integral from ¹⁄₃ to 1 allows a Riemann sum as well, coincidentally again producing . Replacing ¹⁄₃ by an arbitrary positive value s (with s < 1) is equally safe, giving . This, too, has a finite limit as s goes to zero, namely . Combining the limits of the two fragments, the result of this improper integral is

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 does not converge; and over the unbounded interval 1 to ∞ the integral of 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

But the similar integral

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 is defined as either the finite number where b is a point at which the behavior of the function f is such that for any a < b and for any c > b (one sign is + and the other is −). or...

Multiple integration

Main article: Multiple integral

Double integral as volume under a surface.

Integrals can be taken over regions other than intervals. In general, an integral over a set E of a function f is written: To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

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 other words, the integral can be calculated by integrating one coordinate at a time. This article is about vectors that have a particular relation to the spatial coordinates. ... It has been suggested that A counterexample related to Fubinis theorem be merged into this article or section. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

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. (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. This article is about the physical quantity. ... For other uses, see Volume (disambiguation). ... In mathematics, the domain of a function is the set of all input values to the function. ... For other uses, see Fourth dimension (disambiguation). ...

For example, the volume of the parallelepiped of sides 4 × 6 × 5 may be obtained in two ways: In geometry, a parallelepiped (now usually pronounced , traditionally[1] in accordance with its etymology in Greek παραλληλ-επίπεδον, a body having parallel planes) is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ...

• By the double integral

of the function f(x, y) = 5 calculated in the region D in the xy-plane which is the base of the parallelepiped.

• By the triple integral

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 is equal to f, i. ...

Line integrals

Main article: Line integral

A line integral sums together elements along a curve.

The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces. Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with vector fields. This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Vector field given by vectors of the form (−y, x) 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. Various different line integrals are in use. In the case of a closed curve it is also called a contour integral. This article is about functions in mathematics. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that work is equal to force multiplied by distance may be expressed (in terms of vector quantities) as: In mathematics and physics, a scalar field associates a scalar to every point in space. ... Vector field given by vectors of the form (−y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space. ... 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 additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... The differential dy In calculus, a differential is an infinitesimally small change in a variable. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In physics, mechanical work is the amount of energy transferred by a force. ... For other uses, see Force (disambiguation). ...

;

which is paralleled by the line integral:

;

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.

A surface integral is a definite integral taken over a surface (which may be a curved set in space); it can be thought of as the double integral analog of the line integral. The function to be integrated may be a scalar field or a vector field. The value of the surface integral is the sum of the field at all points on the surface. This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums. In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ... Image File history File links Size of this preview: 800 × 550 pixelsFull resolution (1164 × 800 pixel, file size: 79 KB, MIME type: image/png) % An illustration of the surface integral. ... Image File history File links Size of this preview: 800 × 550 pixelsFull resolution (1164 × 800 pixel, file size: 79 KB, MIME type: image/png) % An illustration of the surface integral. ... An open surface with X-, Y-, and Z-contours shown. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... This article is about the idea of space. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... In mathematics and physics, a scalar field associates a scalar to every point in space. ... Vector field given by vectors of the form (−y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space. ...

For an example of applications of surface integrals, consider a vector field v on a surface S; that is, for each point x in S, v(x) is a vector. Imagine that we have a fluid flowing through S, such that v(x) determines the velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through S in unit amount of time. To find the flux, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, which we integrate over the surface: flux in science and mathematics. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

.

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. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In physics, a classical theory usually refers to a theory that does not obey the principles of quantum mechanics (classical theory vs. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ...

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. 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. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... Multivariable calculus is the extension of calculus in one variable to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ...

We initially work in an open set in Rⁿ. A 0-form is defined to be a smooth function f. When we integrate a function f over an m-dimensional subspace S of Rⁿ, we write it as In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... This article is about functions in mathematics. ... 2-dimensional renderings (ie. ...

(The superscripts are indices, not exponents.) We can consider dx¹ through dxⁿ to be formal objects themselves, rather than tags appended to make integrals look like Riemann sums. Alternatively, we can view them as covectors, and thus a measure of "density" (hence integrable in a general sense). We call the dx¹, …,dxⁿ basic 1-forms. In mathematics, a Riemann sum is a method for approximating the values of integrals. ... 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 physical sizes). ... 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 In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. ...

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. In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...

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. 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 mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

In addition to the wedge product, there is also the exterior derivative operator d. 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: In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

with extension to general k-forms occurring linearly.

This more general approach allows for a more natural coordinate-free approach to integration on manifolds. It also allows for a natural generalisation of the fundamental theorem of calculus, called Stokes' theorem, which we may state as On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

where ω is a general k-form, and ∂Ω denotes the boundary of the region Ω. 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. 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. Similarly, using 2-forms, and 3-forms and Hodge duality, we can arrive at Stokes' theorem and the divergence theorem. In this way we can see that differential forms provide a powerful unifying view of integration. In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special two-dimensional case of... In mathematics, the Hodge star operator or Hodge dual is a signficant linear map introduced in general by W. V. D. Hodge. ... Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the surface. ...

Methods and applications

Computing integrals

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

1. Choose a function f(x) and an interval [a, b].
2. Find an antiderivative of f, that is, a function F such that F' = f.
3. By the fundamental theorem of calculus, provided the integrand and integral have no singularities on the path of integration,

   

4. Therefore the value of the integral is 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 well-behaved in some particular way, such as differentiability. ...

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:

• Integration by substitution
• Integration by parts
• Integration by trigonometric substitution
• Integration by partial fractions
• Integration by reduction formulae

Even if these techniques fail, it may still be possible to evaluate a given integral. The next most common technique is residue calculus, whilst for nonelementary integrals Taylor series can sometimes be used to find the antiderivative. There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral. In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. ... 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 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. ... Series expansion redirects here. ... In functional analysis, Parsevals identity, also known as Parsevals equality, is the Pythagorean theorem for inner-product spaces. ... The integral of any Gaussian function (named after Carl Friedrich Gauss) is quickly reducible to the Gaussian integral This integral cannot be computed by elementary means since the function has no simple antiderivative. ...

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 figure around some straight line (the axis) that lies on the same plane. ... In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ... Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ...

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 List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of...

Symbolic algorithms

Main article: Symbolic integration

Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive tables of integrals have been compiled and published over the years for this purpose. With the spread of computers, many professionals, educators, and students have turned to computer algebra systems that are specifically designed to perform difficult or tedious tasks, including integration. Symbolic integration presents a special challenge in the development of such systems. Symbolic integration is the application of computer software to solving problems in mathematics of find the integral of an expression, but finding an expression rather than a value. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... This article is about the machine. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ...

A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather innocently looking function simply 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. Differential Galois theory provides general criteria that allow one to determine whether the antiderivative of an elementary function is elementary. 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. In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... The exponential function is one of the most important functions in mathematics. ... Look up logarithm in Wiktionary, the free dictionary. ... Sine redirects here. ... In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ... In mathematics, several functions are important enough to deserve their own name. ... // In mathematics, the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. ... The Risch algorithm is an algorithm for the calculus operation of indefinite integration (i. ... For other uses, see Mathematica (disambiguation). ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... In mathematics, an nth 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). Extending the Risch-Norman algorithm so that it includes these functions is possible but challenging. In mathematics, there is a theory or theories of special functions, particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In mathematics, the associated Legendre polynomials, named after Adrien-Marie Legendre, are defined by: These differ from the Legendre polynomials. ... In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ... The Gamma function along part of the real axis In mathematics, the Gamma function (represented by the capitalized Greek letter Γ) is an extension of the factorial function to real and complex numbers. ...

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 does this present 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. 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. 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. Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... This article is about the machine. ... ENIAC ENIAC, short for Electronic Numerical Integrator And Computer,[1] was the first large-scale, electronic, digital computer capable of being reprogrammed to solve a full range of computing problems,[2] although earlier computers had been built with some of these properties. ...

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

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 0.25, 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. Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1260 × 1260 pixel, file size: 90 KB, MIME type: image/png) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1260 × 1260 pixel, file size: 90 KB, MIME type: image/png) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In mathematics, the rectangle method of integral calculus uses an approximation to a definite integral, made by finding the area of a series of rectangles. ...

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. 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. The function f(x) (in blue) is approximated by a linear function (in red). ...

Romberg's method builds on the trapezoid method to great effect. 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). With this method a numerically exact answer here requires only four pieces (five function values)! The Lagrange polynomial interpolating {h_k,T(h_k)}_k=0…2 = {(4.00,6.128), (2.00,4.352), (1.00,3.908)} is 3.76+0.148h², producing the extrapolated value 3.76 at h = 0. In numerical analysis, Rombergs method generates a triangular array consisting of numerical estimates of the definite integral by using Richardson extrapolation repeatedly on the trapezium rule. ... For other uses, see Interpolation (disambiguation). ... In numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation polynomial for a given set of data points in the Lagrange form. ...

Gaussian quadrature often requires noticeably less work for superior accuracy. In this example, it can compute the function values at just two x positions, ±²⁄_√3, then double each value and sum to get the numerically exact answer. The explanation for this dramatic success lies in 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.) In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. ...

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

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. Adaptive quadrature is a process in which the integral of a function is approximated using static quadrature rules on adaptively refined subintervals of the integration domain. ...

This brief introduction omits higher-dimensional integrals (for example, area and volume calculations), where alternatives such as Monte Carlo integration have great importance. This article describes multidimensional Monte Carlo integration. ...

A calculus text is no substitute for numerical analysis, but the reverse is also true. Even the best adaptive numerical code sometimes requires a user to help with the more demanding integrals. For example, improper integrals may require a change of variable or methods that can avoid infinite function values; and known properties like symmetry and periodicity may provide critical leverage.

See also

Mathematics Portal

Image File history File links Portal. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... To meet Wikipedias quality standards, this article or section may require cleanup. ... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ... Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ... 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 â„›, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. ... In mathematics, a Riemann sum is a method for approximating the values of integrals. ... Differentiation under the integral sign is an operation in the mathematical field of calculus. ... Product integrals are a multiplicative version of standard integrals of infinitesimal calculus. ...

References

• Apostol, Tom M. (1967), Calculus, Vol. 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 . In particular chapters III and IV.
• Burton, David M. (2005), The History of Mathematics: An Introduction (6^th ed.), McGraw-Hill, p. p. 359, ISBN 978-0-07-305189-5 
• Cajori, Florian (1929), A History Of Mathematical Notations Volume II, Open Court Publishing, pp. 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. (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. §231, <http://books.google.com/books?id=TDQJAAAAIAAJ> 
  Available in translation as Fourier, Joseph (1878), The analytical theory of heat, Cambridge University Press, pp. pp. 200–201, <http://www.archive.org/details/analyticaltheory00fourrich> 
• Heath, T. L., ed. (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. 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., 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.), McGraw-Hill, ISBN 978-0-07-100276-9 
• Saks, Stanisław (1964), Theory of the integral (English translation by L. 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 .
• W3C (2006), Arabic mathematical notation, <http://www.w3.org/TR/arabic-math/> 

Tom Apostol is an analytic number theorist who teaches at California Institute of Technology. ... John Wiley & Sons, Inc. ... This article is about the group of mathematicians named Nicolas Bourbaki. ... Springer Science+Business Media or Springer (IPA: ) is a worldwide publishing company based in Germany which focuses on academic journals and books in the fields of science, technology, mathematics, and medicine. ... The McGraw-Hill Companies, Inc. ... Florian Cajori at Colorado College Florian Cajori was born February 28, 1859 in St Aignan (near Thusis), Graubünden, Switzerland. ... The Open Court Publishing Company is a publisher with offices in Chicago and La Salle, Illinois. ... Germund Dahlquist (January 16, 1925 Uppsala - February 8, 2005 Stockholm) was a Swedish mathematician known primarily for his early contributions to the theory of numerical analysis as applied to differential equations. ... For the country formerly called Siam see Thailand SIAM is an acronym for Society for Industrial and Applied Mathematics. ... John Wiley & Sons, Inc. ... Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... The headquarters of the Cambridge University Press, in Trumpington Street, Cambridge. ... Sir Thomas Little Heath (October 5, 1861 - March 16, 1940) was a mathematician, classical scholar, historian of ancient Greek mathematics, and translator. ... Dover Publications is a book publisher founded in 1941. ... The headquarters of the Cambridge University Press, in Trumpington Street, Cambridge. ... The American Mathematical Society (AMS) is dedicated to the interests of mathematical research and education, which it does with various publications and conferences as well as annual monetary awards to mathematicians. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ... Cleve Barry Moler is a mathematician and computer programmer specializing in numerical analysis. ... Pearson can mean Pearson PLC the media conglomerate. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Walter Rudin Walter Rudin is an American mathematician, formerly a professor of mathematics at the University of Wisconsin, Madison. ... The McGraw-Hill Companies, Inc. ... StanisÅ‚aw Saks (30 December 1897 – 23 November 1942) was a Ukrainian mathematician of Polish and Jewish ethnicity. ... Springer Science+Business Media or Springer (IPA: ) is a worldwide publishing company based in Germany which focuses on academic journals and books in the fields of science, technology, mathematics, and medicine. ...

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.S. Wang, Evaluation of Definite Integrals by Symbolic Manipulation (1972) - a cookbook of definite integral techniques
• Definite Integrals

Wolfram Research is part of the Wolfram Group which consists of four companies: Wolfram Research Inc. ... Maxima is a free computer algebra system based on a 1982 version of Macsyma. ...

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