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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an integral assigns numbers to functions in a way that describes
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
,
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an o ...
,
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
, and other concepts that arise by combining
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an o ...
of the region in the plane that is bounded by the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of a given function between two points in the
real line In elementary mathematics, a number line is a picture of a graduated straight line Line most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dim ...
. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function (mathematics), function is a differentiable function whose derivative is equal to the original function . This ca ...
, a function whose derivative is the given function. In this case, they are called indefinite integrals. The
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known. Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
and
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
width.
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mostly ...
later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a
curvilinear In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the fiel ...
region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century,
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are con ...
generalized Riemann's formulation by introducing what is now referred to as the
Lebesgue integral In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the -axis. The Lebesgue integral, ...
; it is more robust than Riemann's in the sense that a wider class of functions are Lebesgue-integrable. Integrals may be generalized depending on the type of the function as well as the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Dom ...
over which the integration is performed. For example, a
line integral In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integra ...
is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to Integral, integration over surface (differential geometry), surfaces. It can be thought of as the double integral analogue of th ...
, the curve is replaced by a piece of a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of Visual perception, sight ...
in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
.


History


Pre-calculus integration

The first documented systematic technique capable of determining integrals is the
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by Inscribed figure, inscribing inside it a sequence of polygons whose areas limit (mathematics), converge to the area of the containing shape. If the sequence is correctly c ...
of the
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek Dark ...
astronomer Eudoxus (''ca.'' 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. This method was further developed and employed by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, ...
in the 3rd century BC and used to calculate the
area of a circle In geometry, the area enclosed by a circle of radius is . Here the Greek letter pi, represents the constant (mathematics), constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159. One method of deriving ...
, the
surface area The surface area of a solid object is a measure of the total area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a s ...
and
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, area of an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
, the area under a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
, the volume of a segment of a
paraboloid In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of ...
of revolution, the volume of a segment of a
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface (mathematics), surface generated by rotating a hyperbola around one of its Hyperbola#Nomenclature and features, principal axes. A hyperboloid is th ...
of revolution, and the area of a
spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ...
around the 3rd century AD by
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu (The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state o ...
, who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians
Zu Chongzhi Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3.1 ...
and Zu Geng to find the volume of a sphere. In the Middle East, Hasan Ibn al-Haytham, Latinized as
Alhazen Ḥasan Ibn al-Haytham, Latinization of names, Latinized as Alhazen (; full name ; ), was a medieval Mathematics in medieval Islam, mathematician, Astronomy in the medieval Islamic world, astronomer, and Physics in the medieval Islamic world, ...
( AD) derived a formula for the sum of
fourth power In arithmetic Arithmetic () is an elementary part of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities ...
s. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a
paraboloid In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of ...
. The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of Cavalieri with his
method of Indivisibles In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...
, and work by
Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French people, French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In pa ...
, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of up to degree in Cavalieri's quadrature formula. Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
. Wallis generalized Cavalieri's method, computing integrals of to a general power, including negative powers and fractional powers.


Leibniz and Newton

The major advance in integration came in the 17th century with the independent discovery of the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
by
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
and Newton. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, whose notation for integrals is drawn directly from the work of Leibniz.


Formalization

While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of
rigour Rigour (British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the English lan ...
.
Bishop Berkeley George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley (Bishop of Cloyne of the Anglican Church of Ireland) – was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immateri ...
memorably attacked the vanishing increments used by Newton, calling them " ghosts of departed quantities". Calculus acquired a firmer footing with the development of
limits Limit or Limits may refer to: Arts and media * Limit (manga), ''Limit'' (manga), a manga by Keiko Suenobu * Limit (film), ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 201 ...
. 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—particularly in the context of
Fourier analysis In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
—to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in
measure theory In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
(a subfield of
real analysis In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the
hyperreal number In mathematics, the system of hyperreal numbers is a way of treating Infinity, infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an Field extension, extension of the real numbe ...
system.


Historical notation

The notation for the indefinite integral was introduced by
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
in 1675. He adapted the integral symbol, ∫, from the letter ''ſ'' (
long s The long s , also known as the medial s or initial s, is an archaism, archaic form of the lowercase letter . It replaced the single ''s'', or one or both of the letters ''s'' in a 'double ''s sequence (e.g., "ſinfulneſs" for "sinfulness" ...
), standing for ''summa'' (written as ''ſumma''; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by
Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French people, French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier an ...
in ''Mémoires'' of the French Academy around 1819–20, reprinted in his book of 1822.
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
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 are used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.


First use of the term

The term was first printed in Latin by
Jacob Bernoulli Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Leib ...
in 1690: "Ergo et horum Integralia aequantur".


Terminology and notation

In general, the integral of a
real-valued function In mathematics, a real-valued function is a function whose values are real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, durati ...
with respect to a real variable on an interval is written as :\int_^ f(x) \,dx. The integral sign represents integration. The symbol , called the differential of the variable , indicates that the variable of integration is . The function is called the integrand, the points and are called the limits (or bounds) of integration, and the integral is said to be over the interval , called the interval of integration.. A function is said to be if its integral over its domain is finite. If limits are specified, the integral is called a definite integral. When the limits are omitted, as in : \int f(x) \,dx, the integral is called an indefinite integral, which represents a class of functions (the
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function (mathematics), function is a differentiable function whose derivative is equal to the original function . This ca ...
) whose derivative is the integrand. The
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
relates the evaluation of definite integrals to indefinite integrals. There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article). In advanced settings, it is not uncommon to leave out when only the simple Riemann integral is being used, or the exact type of integral is immaterial. For instance, one might write \int_a^b (c_1f+c_2g) = c_1\int_a^b f + c_2\int_a^b g to express the linearity of the integral, a property shared by the Riemann integral and all generalizations thereof.


Interpretations

Integrals appear in many practical situations. For instance, from the length, width and depth of a swimming pool which is rectangular with a flat bottom, one can determine the volume of water it can contain, the area of its surface, and the length of its edge. But if it is oval with a rounded bottom, integrals are required to find exact and rigorous values for these quantities. In each case, one may divide the sought quantity into infinitely many
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
pieces, then sum the pieces to achieve an accurate approximation. For example, to find the area of the region bounded by the graph of the function between and , one can cross the interval in five steps (), then fill a rectangle using the right end height of each piece (thus ) and sum their areas to get an approximation of :\textstyle \sqrt\left(\frac-0\right)+\sqrt\left(\frac-\frac\right)+\cdots+\sqrt\left(\frac-\frac\right)\approx 0.7497, which is larger than the exact value. Alternatively, when replacing these subintervals by ones with the left end height of each piece, the approximation one gets is too low: with twelve such subintervals the approximated area is only 0.6203. However, when the number of pieces increase to infinity, it will reach a limit which is the exact value of the area sought (in this case, ). One writes :\int_^ \sqrt \,dx = \frac, which means is the result of a weighted sum of function values, , multiplied by the infinitesimal step widths, denoted by , on the interval .


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 are Riemann integrals and Lebesgue integrals.


Riemann integral

The Riemann integral is defined in terms of
Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
s of functions with respect to ''tagged partitions'' of an interval. A tagged partition of a
closed interval In mathematics, a (real) interval is a set (mathematics), set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers ...
on the real line is a finite sequence : a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_ \le t_n \le x_n = b . \,\! This partitions the interval into sub-intervals indexed by , each of which is "tagged" with a distinguished point . A ''Riemann sum'' of a function with respect to such a tagged partition is defined as : \sum_^n f(t_i) \, \Delta_i ; thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the width of sub-interval, . The ''mesh'' of such a tagged partition is the width of the largest sub-interval formed by the partition, . The ''Riemann integral'' of a function over the interval is equal to if: : For all \varepsilon > 0 there exists \delta > 0 such that, for any tagged partition , b/math> with mesh less than \delta, : \left, S - \sum_^n f(t_i) \, \Delta_i \ < \varepsilon. 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.


Lebesgue integral

It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral. Therefore, it is of great importance to have a definition of the integral that allows a wider class of functions to be integrated. Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are con ...
introduced the integral bearing his name, explaining this integral thus in a letter to Paul Montel: As Folland puts it, "To compute the Riemann integral of , one partitions the domain into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of ". The definition of the Lebesgue integral thus begins with a measure, μ. In the simplest case, the
Lebesgue measure In Measure (mathematics), measure theory, a branch of mathematics, the Lebesgue measure, named after france, French mathematician Henri Lebesgue, is the standard way of assigning a measure (mathematics), measure to subsets of ''n''-dimensional Eucl ...
of an interval is its width, , so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals. Using the "partitioning the range of " philosophy, the integral of a non-negative function should be the sum over of the areas between a thin horizontal strip between and . This area is just . Let . The Lebesgue integral of is then defined by : \int f = \int_0^\infty f^*(t)\,dt where the integral on the right is an ordinary improper Riemann integral ( is a strictly decreasing positive function, and therefore has a
well-defined In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
improper Riemann integral). For a suitable class of functions (the
measurable function In mathematics and in particular Mathematical analysis#Measure_theory, measure theory, a measurable function is a function between the underlying sets of two measurable space, measurable spaces that preserves the structure of the spaces: the prei ...
s) this defines the Lebesgue integral. A general measurable function is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of and the -axis is finite: : \int_E , f, \,d\mu < + \infty. In that case, the integral is, as in the Riemannian case, the difference between the area above the -axis and the area below the -axis:. : \int_E f \,d\mu = \int_E f^+ \,d\mu - \int_E f^- \,d\mu where : \begin & f^+(x) &&= \max \ &&= \begin f(x), & \text f(x) > 0, \\ 0, & \text \end\\ & f^-(x) &&= \max \ &&= \begin -f(x), & \text f(x) < 0, \\ 0, & \text \end \end


Other integrals

Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including: * The Darboux integral, which is defined by Darboux sums (restricted Riemann sums) yet is equivalent to the
Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function (mathematics), function on an Interval (mathematics), interval. It was pre ...
. A function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of being easier to define than Riemann integrals. * The Riemann–Stieltjes integral, an extension of the Riemann integral which integrates with respect to a function as opposed to a variable. * The Lebesgue–Stieltjes integral, further developed by Johann Radon, which generalizes both the Riemann–Stieltjes and Lebesgue integrals. * The Daniell integral, which subsumes the Lebesgue integral and Lebesgue–Stieltjes integral without depending on measures. * The Haar integral, used for integration on locally compact topological groups, introduced by Alfréd Haar in 1933. * The Henstock–Kurzweil integral, variously defined by Arnaud Denjoy, Oskar Perron, and (most elegantly, as the gauge integral) Jaroslav Kurzweil, and developed by Ralph Henstock. * The Itô integral and Stratonovich integral, which define integration with respect to semimartingales such as
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of Randomness, random fluctuations in a particle's po ...
. * The Young integral, which is a kind of Riemann–Stieltjes integral with respect to certain functions of unbounded variation. * The rough path integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both semimartingales and processes such as the
fractional Brownian motion In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a Continuous-time stoch ...
. * The Choquet integral, a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. * The Bochner integral, an extension of the Lebesgue integral to a more general class of functions, namely, those with a domain that is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a Complete metric space, complete normed vector space. Thus, a Banach space is a vector space with a Metric (mathematics), metric that allows the computation ...
.


Properties


Linearity

The collection of Riemann-integrable functions on a closed interval forms a
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
under the operations of pointwise addition and multiplication by a scalar, and the operation of integration : f \mapsto \int_a^b f(x) \; dx is a
linear functional In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
on this vector space. Thus, the collection of integrable functions is closed under taking linear combinations, and the integral of a linear combination is the linear combination of the integrals:. : \int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \, Similarly, the set of real-valued Lebesgue-integrable functions on a given
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the σ-algebra, -algebra) and the me ...
with measure is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral : f\mapsto \int_E f \, d\mu is a linear functional on this vector space, so that: : \int_E (\alpha f + \beta g) \, d\mu = \alpha \int_E f \, d\mu + \beta \int_E g \, d\mu. More generally, consider the vector space of all
measurable function In mathematics and in particular Mathematical analysis#Measure_theory, measure theory, a measurable function is a function between the underlying sets of two measurable space, measurable spaces that preserves the structure of the spaces: the prei ...
s on a measure space , taking values in a
locally compact 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. More precisely, it is a topological space in which ev ...
complete Complete may refer to: Logic * Completeness (logic) * Complete theory, Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, ...
topological vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
over a locally compact topological field . Then one may define an abstract integration map assigning to each function an element of or the symbol , : f\mapsto\int_E f \,d\mu, \, that is compatible with linear combinations. In this situation, the linearity holds for the subspace of functions whose integral is an element of (i.e. "finite"). The most important special cases arise when is , , or a finite extension of the field of
p-adic number In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s, and is a finite-dimensional vector space over , and when and is a complex
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
. Linearity, together with some natural continuity properties and normalization for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of Daniell for the case of real-valued functions on a set , generalized by
Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally in ...
to functions with values in a locally compact topological vector space. See for an axiomatic characterization of the integral.


Inequalities

A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval and can be generalized to other notions of integral (Lebesgue and Daniell). * ''Upper and lower bounds.'' An integrable function on , is necessarily bounded on that interval. Thus there are
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s and so that for all in . Since the lower and upper sums of over are therefore bounded by, respectively, and , it follows that m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a). * ''Inequalities between functions.'' If for each in then each of the upper and lower sums of is bounded above by the upper and lower sums, respectively, of . Thus \int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx. This is a generalization of the above inequalities, as is the integral of the constant function with value over . In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if for each in , then \int_a^b f(x) \, dx < \int_a^b g(x) \, dx. * ''Subintervals.'' If is a subinterval of and is non-negative for all , then \int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx. * ''Products and absolute values of functions.'' If and are two functions, then we may consider their
pointwise product In mathematics, the pointwise product of two function (mathematics), functions is another function, obtained by multiplying the Image (mathematics), images of the two functions at each value in the domain of a function, domain. If and are both ...
s and powers, and
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign (mathematics), sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative number, negative (in which cas ...
s: (fg)(x)= f(x) g(x), \; f^2 (x) = (f(x))^2, \; , f, (x) = , f(x), . If is Riemann-integrable on then the same is true for , and \left, \int_a^b f(x) \, dx \ \leq \int_a^b , f(x) , \, dx. Moreover, if and are both Riemann-integrable then is also Riemann-integrable, and \left( \int_a^b (fg)(x) \, dx \right)^2 \leq \left( \int_a^b f(x)^2 \, dx \right) \left( \int_a^b g(x)^2 \, dx \right). This inequality, known as the
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used Inequality (mathematics), inequalities in mathematics. The inequality for sums was published by . The c ...
, plays a prominent role in
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
theory, where the left hand side is interpreted as the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
of two square-integrable functions and on the interval . * ''Hölder's inequality''.. Suppose that and are two real numbers, with , and and are two Riemann-integrable functions. Then the functions and are also integrable and the following Hölder's inequality holds: \left, \int f(x)g(x)\,dx\ \leq \left(\int \left, f(x)\^p\,dx \right)^ \left(\int\left, g(x)\^q\,dx\right)^. For , Hölder's inequality becomes the Cauchy–Schwarz inequality. * ''Minkowski inequality''. Suppose that is a real number and and are Riemann-integrable functions. Then and are also Riemann-integrable and the following
Minkowski inequality In mathematical analysis, the Minkowski inequality establishes that the Lp space, L''p'' spaces are normed vector spaces. Let ''S'' be a measure space, let and let ''f'' and ''g'' be elements of L''p''(''S''). Then is in L''p''(''S''), and we ha ...
holds: \left(\int \left, f(x)+g(x)\^p\,dx \right)^ \leq \left(\int \left, f(x)\^p\,dx \right)^ + \left(\int \left, g(x)\^p\,dx \right)^. An analogue of this inequality for Lebesgue integral is used in construction of Lp spaces.


Conventions

In this section, is a
real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object A mathematical object is an abstract concept arising in mathematics Mathematics is an area ...
Riemann-integrable function. The integral : \int_a^b f(x) \, dx over an interval is defined if . This means that the upper and lower sums of the function are evaluated on a partition whose values are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating within intervals where an interval with a higher index lies to the right of one with a lower index. The values and , the end-points of the interval, are called the limits of integration of . Integrals can also be defined if :'''' :\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx. With , this implies: :\int_a^a f(x) \, dx = 0. The first convention is necessary in consideration of taking integrals over subintervals of ; the second says that an integral taken over a degenerate interval, or a point, should be
zero 0 (zero) is a number, and the numerical digit used to represent that number in numeral system, numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. A ...
. One reason for the first convention is that the integrability of on an interval implies that is integrable on any subinterval , but in particular integrals have the property that if is any element of , then:'''' : \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx. With the first convention, the resulting relation : \begin \int_a^c f(x) \, dx &= \int_a^b f(x) \, dx - \int_c^b f(x) \, dx \\ & = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx \end is then well-defined for any cyclic permutation of , , and .


Fundamental theorem of calculus

The ''fundamental theorem of calculus'' is the statement that differentiation and integration are inverse operations: if a
continuous function In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value ...
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.


First theorem

Let be a continuous real-valued function defined on a
closed interval In mathematics, a (real) interval is a set (mathematics), set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers ...
. Let be the function defined, for all in , by : F(x) = \int_a^x f(t)\, dt. Then, is continuous on , differentiable on the open interval , and : F'(x) = f(x) for all in .


Second theorem

Let be a real-valued function defined on a
closed interval In mathematics, a (real) interval is a set (mathematics), set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers ...
[] that admits an
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function (mathematics), function is a differentiable function whose derivative is equal to the original function . This ca ...
on . That is, and are functions such that for all in , : f(x) = F'(x). If is integrable on then : \int_a^b f(x)\,dx = F(b) - F(a).


Extensions


Improper integrals

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 In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
of proper
Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function (mathematics), function on an Interval (mathematics), interval. It was pre ...
s on progressively larger intervals. If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity: : \int_a^\infty f(x)\,dx = \lim_ \int_a^b f(x)\,dx. If the integrand is only defined or finite on a half-open interval, for instance , then again a limit may provide a finite result: : \int_a^b f(x)\,dx = \lim_ \int_^ f(x)\,dx. That is, the improper integral is the limit of proper integrals as one endpoint of the interval of integration approaches either a specified
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
, or , or . In more complicated cases, limits are required at both endpoints, or at interior points.


Multiple integration

Just as the definite integral of a positive function of one variable represents the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an o ...
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 Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of the region between the surface defined by the function and the plane that contains its domain. For example, a function in two dimensions depends on two real variables, ''x'' and ''y'', and the integral of a function ''f'' over the rectangle ''R'' given as the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two Set (mathematics), sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notatio ...
of two intervals R= ,btimes ,d/math> can be written : \int_R f(x,y)\,dA where the differential indicates that integration is taken with respect to area. This double integral can be defined using
Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
s, and represents the (signed) volume under the graph of over the domain ''R''.. Under suitable conditions (e.g., if ''f'' is continuous),
Fubini's theorem In mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, i ...
states that this integral can be expressed as an equivalent iterated integral : \int_a^b\left int_c^d f(x,y)\,dy\right,dx. This reduces the problem of computing a double integral to computing one-dimensional integrals. Because of this, another notation for the integral over ''R'' uses a double integral sign: : \iint_R f(x,y) \, dA. Integration over more general domains is possible. The integral of a function ''f'', with respect to volume, over an ''n-''dimensional region ''D'' of \mathbb^n is denoted by symbols such as: : \int_D f(\mathbf x) d^n\mathbf x \ = \int_D f\,dV.


Line integrals and surface integrals

The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces inside higher-dimensional spaces. Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with vector fields. A ''line integral'' (sometimes called a ''path integral'') is an integral where the function to be integrated is evaluated along a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
. Various different line integrals are in use. In the case of a closed curve it is also called a ''contour integral''. The function to be integrated may be a
scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
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 ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
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 In physics, a force is an influence that can change the motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to accelerate. Force can ...
, , multiplied by displacement, , may be expressed (in terms of vector quantities) as: : W=\mathbf F\cdot\mathbf s. For an object moving along a path in a vector field such as an
electric field An electric field (sometimes E-field) is the field (physics), physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the ...
or
gravitational field In physics, a gravitational field is a scientific model, model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational Field (physics), field is us ...
, the total work done by the field on the object is obtained by summing up the differential work done in moving from to . This gives the line integral : W=\int_C \mathbf F\cdot d\mathbf s. A ''surface integral'' generalizes double integrals to integration over a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of Visual perception, sight ...
(which may be a curved set in
space Space is the boundless Three-dimensional space, three-dimensional extent in which Physical body, objects and events have relative position (geometry), position and direction (geometry), direction. In classical physics, physical space is often ...
); it can be thought of as the double integral analog of the
line integral In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integra ...
. The function to be integrated may be a
scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
or a vector field. The value of the surface integral is the sum of the field at all points on the surface. This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums. For an example of applications of surface integrals, consider a vector field on a surface ; that is, for each point in , is a vector. Imagine that a fluid flows through , such that determines the velocity of the fluid at . The
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
is defined as the quantity of fluid flowing through in unit amount of time. To find the flux, one need to take the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar (mathematics), scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidea ...
of with the unit
surface normal In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field ...
to at each point, which will give a scalar field, which is integrated over the surface: : \int_S \cdot \,d. 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 In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
.


Contour integrals

In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the integrand is a complex-valued function of a complex variable instead of a real function of a real variable . When a complex function is integrated along a curve \gamma in the complex plane, the integral is denoted as follows : \int_\gamma f(z)\,dz. This is known as a
contour integral In the mathematical field of complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex nu ...
.


Integrals of differential forms

A
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
is a mathematical concept in the fields of
multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change ...
,
differential topology In mathematics, differential topology is the field dealing with the topology, topological properties and smooth structure, smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of dif ...
, and
tensor In mathematics, a tensor is an mathematical object, algebraic object that describes a Multilinear map, multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as Vect ...
s. Differential forms are organized by degree. For example, a one-form is a weighted sum of the differentials of the coordinates, such as: : E(x,y,z)\,dx + F(x,y,z)\,dy + G(x,y,z)\, dz where ''E'', ''F'', ''G'' are functions in three dimensions. A differential one-form can be integrated over an oriented path, and the resulting integral is just another way of writing a line integral. Here the basic differentials ''dx'', ''dy'', ''dz'' measure infinitesimal oriented lengths parallel to the three coordinate axes. A differential two-form is a sum of the form : G(x,y,z) \, dx\wedge dy + E(x,y,z) \, dy\wedge dz + F(x,y,z) \, dz\wedge dx. Here the basic two-forms dx\wedge dy, dz\wedge dx, dy\wedge dz measure oriented areas parallel to the coordinate two-planes. The symbol \wedge denotes the wedge product, which is similar to the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two Euclidean vector, vectors in a three-dimensional Orientation (vector space), oriented E ...
in the sense that the wedge product of two forms representing oriented lengths represents an oriented area. A two-form can be integrated over an oriented surface, and the resulting integral is equivalent to the surface integral giving the flux of E\mathbf i+F\mathbf j+G\mathbf k. Unlike the cross product, and the three-dimensional vector calculus, the wedge product and the calculus of differential forms makes sense in arbitrary dimension and on more general manifolds (curves, surfaces, and their higher-dimensional analogs). The
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The ...
plays the role of the
gradient In vector calculus, the gradient of a scalar-valued function, scalar-valued differentiable function of Function of several variables, several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "d ...
and
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (computing), library (libcurl) and command-line tool (curl) for transferring data using various network Protocol (computing), protocols. The name sta ...
of vector calculus, and
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theoremNagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, ...
simultaneously generalizes the three theorems of vector calculus: the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface (mathematics), surface to the ''divergence'' o ...
,
Green's theorem In vector calculus, Green's theorem relates a line integral In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral ...
, and the Kelvin-Stokes theorem.


Summations

The discrete equivalent of integration is
summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: function (mathematics), fu ...
. Summations and integrals can be put on the same foundations using the theory of
Lebesgue integral In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the -axis. The Lebesgue integral, ...
s or time-scale calculus.


Functional integrals

An integration that is performed not over a variable (or, in physics, over a space or time dimension), but over a space of functions, is referred to as a functional integral.


Applications

Integrals are used extensively in many areas. For example, in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, integrals are used to determine the probability of some
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
falling within a certain range. Moreover, the integral under an entire
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...
must equal 1, which provides a test of whether a function with no negative values could be a density function or not. Integrals can be used for computing the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an o ...
of a two-dimensional region that has a curved boundary, as well as computing the volume of a three-dimensional object that has a curved boundary. The area of a two-dimensional region can be calculated using the aforementioned definite integral. The volume of a three-dimensional object such as a disc or washer can be computed by disc integration using the equation for the volume of a cylinder, \pi r^2 h , where r is the radius. In the case of a simple disc created by rotating a curve about the -axis, the radius is given by , and its height is the differential . Using an integral with bounds and , the volume of the disc is equal to:\pi \int_a^b f^2 (x) \, dx.Integrals are also used in physics, in areas like
kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
to find quantities like
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
,
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
, and
velocity Velocity is the directional derivative, directional speed of an physical object, object in motion as an indication of its time derivative, rate of change in position (vector), position as observed from a particular frame of reference and as m ...
. For example, in rectilinear motion, the displacement of an object over the time interval ,b/math> is given by: : x(b)-x(a) = \int_a^b v(t) \,dt, where v(t) is the velocity expressed as a function of time. The work done by a force F(x) (given as a function of position) from an initial position A to a final position B is: : W_ = \int_A^B F(x)\,dx. Integrals are also used in
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, where thermodynamic integration is used to calculate the difference in free energy between two given states.


Computation


Analytical

The most basic technique for computing definite integrals of one real variable is based on the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
. Let be the function of to be integrated over a given interval . Then, find an antiderivative of ; that is, a function such that on the interval. Provided the integrand and integral have no singularities on the path of integration, by the fundamental theorem of calculus, :\int_a^b f(x)\,dx=F(b)-F(a). Sometimes it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable. Techniques include
integration by substitution In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for derivative, differentiati ...
,
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral (mathematics), integral of a product (mathematics), product of Function (mathematics), functions in terms o ...
, integration by trigonometric substitution, and integration by partial fractions. Alternative methods exist to compute more complex integrals. Many
nonelementary integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
s can be expanded in a
Taylor series In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution using Meijer G-functions can also be used, assuming that the integrand can be written as a product of Meijer G-functions. 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 The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is \int_^\infty e^\,dx = \s ...
. Computations of volumes of solids of revolution can usually be done with disk integration or shell integration. Specific results which have been worked out by various techniques are collected in the list of integrals.


Symbolic

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 system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s that are specifically designed to perform difficult or tedious tasks, including integration. Symbolic integration has been one of the motivations for the development of the first such systems, like
Macsyma Macsyma (; "Project MAC's SYmbolic MAnipulator") is one of the oldest general-purpose computer algebra systems still in wide use. It was originally developed from 1968 to 1982 at MIT's Project MAC. In 1982, Macsyma was licensed to Symbolics and ...
and
Maple ''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since h ...
. A major mathematical difficulty in symbolic integration is that in many cases, a relatively simple function does not have integrals that can be expressed in closed form involving only
elementary function In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
s, include
rational Rationality is the Quality (philosophy), quality of being guided by or based on reasons. In this regard, a person Action (philosophy), acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong e ...
and
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
functions,
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
,
trigonometric functions In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
and
inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fu ...
, and the operations of multiplication and composition. The Risch algorithm provides a general criterion to determine whether the antiderivative of an elementary function is elementary, and to compute it if it is. However, functions with closed expressions of antiderivatives are the exception, and consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition, and to find the symbolic answer whenever it exists. The Risch algorithm, implemented in
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, Computer algebra, symbolic computation, data manipulation, network analysis, time series analysi ...
,
Maple ''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since h ...
and other
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s, does just that for functions and antiderivatives built from rational functions, radicals, logarithm, and exponential functions. Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the
special functions Special functions are particular mathematical function In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and opera ...
(like the Legendre functions, the hypergeometric function, the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
, the
incomplete gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which ...
and so on). Extending the Risch's algorithm to include such functions is possible but challenging and has been an active research subject. More recently a new approach has emerged, using ''D''-finite functions, which are the solutions of
linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...
s with polynomial coefficients. Most of the elementary and special functions are ''D''-finite, and the integral of a ''D''-finite function is also a ''D''-finite function. This provides an algorithm to express the antiderivative of a ''D''-finite function as the solution of a differential equation. This theory also allows one to compute the definite integral of a ''D''-function as the sum of a series given by the first coefficients, and provides an algorithm to compute any coefficient.


Numerical

Definite integrals may be approximated using several methods of
numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equati ...
. The rectangle method relies on dividing the region under the function into a series of rectangles corresponding to function values and multiplies by the step width to find the sum. A better approach, the
trapezoidal rule In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule; see Trapezoid for more information on terminology) is a technique for approximating the integral, definite integral. \int_a^b f(x) \, dx. The trapezoidal rul ...
, replaces the rectangles used in a Riemann sum with trapezoids. The trapezoidal rule weights the first and last values by one half, then multiplies by the step width to obtain a better approximation. The idea behind the trapezoidal rule, that more accurate approximations to the function yield better approximations to the integral, can be carried further: Simpson's rule approximates the integrand by a piecewise quadratic function. Riemann sums, the trapezoidal rule, and Simpson's rule are examples of a family of quadrature rules called the Newton–Cotes formulas. The degree Newton–Cotes quadrature rule approximates the polynomial on each subinterval by a degree ' polynomial. This polynomial is chosen to interpolate the values of the function on the interval. Higher degree Newton–Cotes approximations can be more accurate, but they require more function evaluations, and they can suffer from numerical inaccuracy due to
Runge's phenomenon In the mathematics, mathematical field of numerical analysis, Runge's phenomenon () is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced i ...
. One solution to this problem is
Clenshaw–Curtis quadrature Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the Integrand#Terminology and notation, integrand in terms of Chebyshev polynomials. Equivalently, they em ...
, in which the integrand is approximated by expanding it in terms of
Chebyshev polynomials The Chebyshev polynomials are two sequences of polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coef ...
. Romberg's method halves the step widths incrementally, giving trapezoid approximations denoted by , , and so on, where is half of . For each new step size, only half the new function values need to be computed; the others carry over from the previous size. It then interpolate a polynomial through the approximations, and extrapolate to . Gaussian quadrature evaluates the function at the roots of a set of
orthogonal polynomials In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
. An -point Gaussian method is exact for polynomials of degree up to . The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives as
Monte Carlo integration In mathematics, Monte Carlo integration is a technique for numerical quadrature, numerical integration using pseudorandomness, random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algori ...
.


Mechanical

The area of an arbitrary two-dimensional shape can be determined using a measuring instrument called planimeter. The volume of irregular objects can be measured with precision by the fluid displaced as the object is submerged.


Geometrical

Area can sometimes be found via geometrical
compass-and-straightedge construction In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...
s of an equivalent
square In Euclidean geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, π/2 radian angles, or right angles). It can also be defined as a rec ...
.


Integration by differentiation

Kempf, Jackson and Morales demonstrated mathematical relations that allow an integral to be calculated by means of differentiation. Their calculus involves the
Dirac delta function In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
and the
partial derivative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
operator \partial_x. This can also be applied to functional integrals, allowing them to be computed by functional differentiation..


Examples


Using the Fundamental Theorem of Calculus

The
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
allows for straightforward calculations of basic functions. \int_^ \sin(x)dx = -\cos(x) \big, ^_ = -\cos(\pi) - (-\cos(0)) = 2


See also

* *


Notes


References


Bibliography

* * * . In particular chapters III and IV. * * * * * *
Available in translation as *
(Originally published by Cambridge University Press, 1897, based on J. L. Heiberg's Greek version.) * * * * * * * * * * * . * * . * *


External links

*
Online Integral Calculator
Wolfram Alpha WolframAlpha ( ) is an answer engine developed by Wolfram Research. It answers factual queries by computing answers from externally sourced data. WolframAlpha was released on May 18, 2009 and is based on Wolfram's earlier product Wolfram Ma ...
.


Online books

* Keisler, H. Jerome
Elementary Calculus: An Approach Using Infinitesimals
University of Wisconsin * Stroyan, K. D.

University of Iowa * Mauch, Sean

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 * Johnson, William Woolsey (1909
Elementary Treatise on Integral Calculus
link from
HathiTrust HathiTrust Digital Library is a large-scale collaborative digital repository, repository of digital content from research libraries including content digitized via Google Books and the Internet Archive digitization initiatives, as well as conten ...
. * 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

at ''Holistic Numerical Methods Institute'' * P. S. Wang
Evaluation of Definite Integrals by Symbolic Manipulation
(1972) — a cookbook of definite integral techniques {{Machine learning evaluation metrics Functions and mappings Linear operators in calculus