View source for Integral

{{Short description|Operation in mathematical calculus}}
{{About|the concept of definite integrals in calculus|the indefinite integral|antiderivative|the set of numbers|integer|other uses|Integral (disambiguation)}}
{{Redirect|Area under the curve|the pharmacology integral|Area under the curve (pharmacokinetics)|the statistics concept|Receiver operating characteristic#Area under the curve}}
[[File:Integral example.svg|thumb|300px|A definite integral of a function can be represented as the [[signed area]] of the region bounded by its graph and the horizontal axis; in the above graph as an example, the integral of <math>f(x)</math> is the yellow (−) area subtracted from the blue (+) area|alt=Definite integral example]]
{{Calculus|Integral}}

In [[mathematics]], an '''integral''' is the continuous analog of a [[Summation|sum]], which is used to calculate [[area|areas]], [[volume|volumes]], and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of [[calculus]],<ref group="lower-alpha">Integral calculus is a very well established mathematical discipline for which there are many sources. See {{Harvnb|Apostol|1967}} and {{Harvnb|Anton|Bivens|Davis|2016}}, for example.</ref> the other being [[Derivative|differentiation]]. Integration was initially used to solve problems in mathematics and [[physics]], such as finding the '''area under a curve''', or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.

A '''definite integral''' computes the [[signed area]] of the region in the plane that is bounded by the [[Graph of a function|graph]] of a given [[Function (mathematics)|function]] between two points in the [[real line]]. Conventionally, areas above the horizontal [[Coordinate axis|axis]] of the plane are positive while areas below are negative. Integrals also refer to the concept of an ''[[antiderivative]]'', a function whose [[derivative]] is the given function; in this case, they are also called ''indefinite integrals''. The [[fundamental theorem of calculus]] relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are [[inverse function|inverse]] operations.

Although methods of calculating areas and volumes dated from [[ancient Greek mathematics]], the principles of integration were formulated independently by [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]] in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of [[infinitesimal]] width. [[Bernhard Riemann]] later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a [[Curvilinear coordinates|curvilinear]] region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, [[Henri Lebesgue]] generalized Riemann's formulation by introducing what is now referred to as the [[Lebesgue integration|Lebesgue integral]]; it is more general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable.

Integrals may be generalized depending on the type of the function as well as the [[Domain (mathematical analysis)|domain]] over which the integration is performed. For example, a [[line integral]] is defined for functions of two or more variables, and the [[Interval (mathematics)|interval]] of integration is replaced by a curve connecting two points in space. In a [[surface integral]], the curve is replaced by a piece of a [[Surface (mathematics)|surface]] in [[three-dimensional space]].

== History ==
{{See also|History of calculus}}

=== Pre-calculus integration ===
The first documented systematic technique capable of determining integrals is the [[method of exhaustion]] of the [[Ancient Greece|ancient Greek]] astronomer [[Eudoxus of Cnidus|Eudoxus]] and philosopher [[Democritus]] (''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.<ref>{{Harvnb|Burton|2011|p=117}}.</ref> This method was further developed and employed by [[Archimedes]] in the 3rd century BC and used to calculate the [[area of a circle]], the [[surface area]] and [[volume]] of a [[sphere]], area of an [[ellipse]], the area under a [[parabola]], the volume of a segment of a [[paraboloid]] of revolution, the volume of a segment of a [[hyperboloid]] of revolution, and the area of a [[spiral]].<ref>{{Harvnb|Heath|2002}}.</ref>

A similar method was independently developed in [[China]] around the 3rd century AD by [[Liu Hui]], who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians [[Zu Chongzhi]] and [[Zu Geng (mathematician)|Zu Geng]] to find the volume of a sphere.<ref>{{harvnb|Katz|2009|pp=201–204}}.</ref>

In the Middle East, Hasan Ibn al-Haytham, Latinized as [[Alhazen]] ({{c.|965|lk=no|1040}}&nbsp;AD) derived a formula for the sum of [[fourth power]]s.<ref>{{harvnb|Katz|2009|pp=284–285}}.</ref> Alhazen determined the equations to calculate the area enclosed by the curve represented by <math>y=x^k</math> (which translates to the integral <math>\int x^k \, dx</math> in contemporary notation), for any given non-negative integer value of <math>k</math>.<ref>{{Cite journal |last=Dennis |first=David |last2=Kreinovich |first2=Vladik |last3=Rump |first3=Siegfried M. |date=1998-05-01 |title=Intervals and the Origins of Calculus |url=https://doi.org/10.1023/A:1009989211143 |journal=Reliable Computing |language=en |volume=4 |issue=2 |pages=191–197 |doi=10.1023/A:1009989211143 |issn=1573-1340|url-access=subscription }}</ref> 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]].<ref>{{harvnb|Katz|2009|pp=305–306}}.</ref>

The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of [[Bonaventura Cavalieri|Cavalieri]] with his [[method of indivisibles]], and work by [[Pierre de Fermat|Fermat]], began to lay the foundations of modern calculus,<ref>{{harvnb|Katz|2009|pp=516–517}}.</ref> with Cavalieri computing the integrals of {{math|''x''<sup>''n''</sup>}} up to degree {{math|''n'' {{=}} 9}} in [[Cavalieri's quadrature formula]].<ref>{{Harvnb|Struik|1986|pp=215–216}}.</ref> The case ''n'' = &minus;1 required the invention of a [[function (mathematics)|function]], the [[hyperbolic logarithm]], achieved by [[quadrature (mathematics)|quadrature]] of the [[hyperbola]] in 1647.

Further steps were made in the early 17th century by [[Isaac Barrow|Barrow]] and [[Evangelista Torricelli|Torricelli]], who provided the first hints of a connection between integration and [[Differential calculus|differentiation]]. Barrow provided the first proof of the [[fundamental theorem of calculus]].<ref>{{harvnb|Katz|2009|pp=536–537}}.</ref> [[John Wallis|Wallis]] generalized Cavalieri's method, computing integrals of {{mvar|x}} to a general power, including negative powers and fractional powers.<ref>{{Harvnb|Burton|2011|pp=385–386}}.</ref>

=== Leibniz and Newton ===
The major advance in integration came in the 17th century with the independent discovery of the [[fundamental theorem of calculus]] by [[Gottfried Wilhelm Leibniz|Leibniz]] and [[Isaac Newton|Newton]].<ref>{{Harvnb|Stillwell|1989|p=131}}.</ref> The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions with continuous domains. This framework eventually became modern [[calculus]], 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 [[Rigor#Mathematical rigour|rigour]]. [[George Berkeley|Bishop Berkeley]] memorably attacked the vanishing increments used by Newton, calling them "[[The Analyst#Content|ghosts of departed quantities]]".<ref>{{harvnb|Katz|2009|pp=628–629}}.</ref> Calculus acquired a firmer footing with the development of [[Limit (mathematics)|limits]]. Integration was first rigorously formalized, using limits, by [[Bernhard Riemann|Riemann]].<ref>{{harvnb|Katz|2009|p=785}}.</ref> 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]]—to which Riemann's definition does not apply, and [[Henri Lebesgue|Lebesgue]] formulated a [[#Lebesgue integral|different definition of integral]], founded in [[Measure (mathematics)|measure theory]] (a subfield of [[real analysis]]). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the [[standard part]] of an infinite Riemann sum, based on the [[hyperreal number]] system.

=== Historical notation ===
The notation for the indefinite integral was introduced by [[Gottfried Wilhelm Leibniz]] in 1675.<ref>{{Harvnb|Burton|2011|loc=p.&nbsp;414}}; {{Harvnb|Leibniz|1899|loc=p.&nbsp;154}}.</ref> He adapted the [[integral symbol]], '''∫''', from the letter ''ſ'' ([[long s]]), standing for ''summa'' (written as ''ſumma''; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by [[Joseph Fourier]] in ''Mémoires'' of the French Academy around 1819–1820, reprinted in his book of 1822.<ref>{{Harvnb|Cajori|1929|loc=pp.&nbsp;249–250}}; {{Harvnb|Fourier|1822|loc=§231}}.</ref>

[[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 {{math|{{overset|'''.'''|''x''}}}} or {{math|''x''′}}, which are used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.<ref>{{Harvnb|Cajori|1929|p=246}}.</ref>

=== First use of the term ===
The term was first printed in Latin by [[Jacob Bernoulli]] in 1690: "Ergo et horum Integralia aequantur".<ref>{{Harvnb|Cajori|1929|p=182}}.</ref>

== Terminology and notation ==
In general, the integral of a [[real-valued function]] {{Math|1=''f''(''x'')}} with respect to a real variable {{Math|1=''x''}} on an interval {{Math|1=[''a'', ''b'']}} is written as
:<math>\int_{a}^{b} f(x) \,dx.</math>
The integral sign {{Math|∫}} represents integration. The symbol {{Math|''dx''}}, called the [[Differential (infinitesimal)|differential]] of the variable {{Math|1=''x''}}, indicates that the variable of integration is {{Math|1=''x''}}.  The function {{Math|1=''f''(''x'')}} is called the ''integrand'', the points {{Math|1=''a''}} and {{Math|1=''b''}} are called the limits (or bounds) of integration, and the integral is said to be over the interval {{Math|1=[''a'', ''b'']}}, called the interval of integration.<ref name=":1">{{Harvnb|Apostol|1967|p=74}}.</ref> 
A function is said to be {{em|integrable}}{{anchor|Integrable|Integrable function}} 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

: <math>\int f(x) \,dx,</math>

the integral is called an indefinite integral, which represents a class of functions (the [[antiderivative]]) whose derivative is the integrand.<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=259}}.</ref> The [[fundamental theorem of calculus]] relates the evaluation of definite integrals to indefinite integrals. There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article).

In advanced settings, it is not uncommon to leave out {{Math|''dx''}} when only the simple [[Riemann integral]] is being used, or the exact type of integral is immaterial. For instance, one might write <math display="inline">\int_a^b (c_1f+c_2g) = c_1\int_a^b f + c_2\int_a^b g </math> to express the linearity of the integral, a property shared by the Riemann integral and all generalizations thereof.<ref>{{Harvnb|Apostol|1967|p=69}}.</ref>

==Interpretations==
[[File:Integral approximations J.svg|thumb|Approximations to integral of {{Math|{{radic|''x''}}}} from 0 to 1, with 5 yellow right endpoint partitions and 10 green left endpoint partitions]]
Integrals appear in many practical situations. For instance, from the length, width and depth of a swimming pool which is rectangular with a flat bottom, one can determine the volume of water it can contain, the area of its surface, and the length of its edge. But if it is oval with a rounded bottom, integrals are required to find exact and rigorous values for these quantities. In each case, one may divide the sought quantity into infinitely many [[infinitesimal]] pieces, then sum the pieces to achieve an accurate approximation.

As another example, to find the area of the region bounded by the graph of the function {{Math|1=''f''(''x'') =}} <math display="inline">\sqrt{x}</math> between {{Math|1=''x'' = 0}} and {{Math|1=''x'' = 1}}, one can divide the interval into five pieces ({{Math|0, 1/5, 2/5, ..., 1}}), then construct rectangles using the right end height of each piece (thus {{Math|{{radic|0}}, {{radic|1/5}}, {{radic|2/5}}, ..., {{radic|1}}}}) and sum their areas to get the approximation

:<math>\textstyle \sqrt{\frac{1}{5}}\left(\frac{1}{5}-0\right)+\sqrt{\frac{2}{5}}\left(\frac{2}{5}-\frac{1}{5}\right)+\cdots+\sqrt{\frac{5}{5}}\left(\frac{5}{5}-\frac{4}{5}\right)\approx 0.7497,</math>
which is larger than the exact value. Alternatively, when replacing these subintervals by ones with the left end height of each piece, the approximation one gets is too low: with twelve such subintervals the approximated area is only 0.6203. However, when the number of pieces increases to infinity, it will reach a limit which is the exact value of the area sought (in this case, {{Math|2/3}}). One writes

:<math>\int_{0}^{1} \sqrt{x} \,dx = \frac{2}{3},</math>
which means {{Math|2/3}} is the result of a weighted sum of function values, {{Math|1={{radic|''x''}}}}, multiplied by the infinitesimal step widths, denoted by {{Math|''dx''}}, on the interval {{Math|1=[0, 1]}}.

{{multiple image
<!-- Essential parameters -->| align = center
| direction = horizontal
| caption_align = center
| width = 300
<!-- Extra parameters -->| header = Darboux sums
| header_align = center
| header_background = 
| footer = 
| footer_align = 
| footer_background = 
| background color = 
| image1 = Riemann Integration and Darboux Upper Sums.gif
| width1 = 300
| caption1 = Darboux upper sums of the function {{math|''y'' {{=}} ''x''<sup>2</sup>}}
| alt1 = Upper Darboux sum example
| image2 = Riemann Integration and Darboux Lower Sums.gif
| width2 = 300
| caption2 = Darboux lower sums of the function {{math|''y'' {{=}} ''x''<sup>2</sup>}}
| alt2 = Lower Darboux sum example
}}

==Formal definitions==
{{multiple image
| align = right
| caption_align = center
| direction = vertical
| width = 200
| image = Integral Riemann sum.png
| alt1 = Riemann integral approximation example
| caption1 = Integral example with irregular partitions (largest marked in red)
| image2 = Riemann sum convergence.png
| alt2 = Riemann sum convergence
| caption2 = Riemann sums converging
}}There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but are also occasionally for pedagogical reasons. The most commonly used definitions are Riemann integrals and Lebesgue integrals.

=== Riemann integral ===
{{Main|Riemann integral}}
The Riemann integral is defined in terms of [[Riemann sum]]s of functions with respect to ''tagged partitions'' of an interval.<ref>{{Harvnb|Anton|Bivens|Davis|2016|pp=286−287}}.</ref> A tagged partition of a [[closed interval]] {{math|[''a'', ''b'']}} on the real line is a finite sequence

: <math> a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!</math>

This partitions the interval {{math|[''a'', ''b'']}} into {{mvar|n}} sub-intervals {{math|[''x''<sub>''i''−1</sub>, ''x''<sub>''i''</sub>]}} indexed by {{mvar|i}}, each of which is "tagged" with a specific point {{math|''t''<sub>''i''</sub> ∈ [''x''<sub>''i''−1</sub>, ''x''<sub>''i''</sub>]}}. A ''Riemann sum'' of a function {{mvar|f}} with respect to such a tagged partition is defined as

: <math>\sum_{i=1}^n f(t_i) \, \Delta_i ; </math>

thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub-interval, {{math|Δ<sub>''i''</sub> {{=}} ''x''<sub>''i''</sub>−''x''<sub>''i''−1</sub>}}. The ''mesh'' of such a tagged partition is the width of the largest sub-interval formed by the partition, {{math|max<sub>''i''{{=}}1...''n''</sub> Δ<sub>''i''</sub>}}. The ''Riemann integral'' of a function {{mvar|f}} over the interval {{math|[''a'', ''b'']}} is equal to {{mvar|S}} if:<ref>{{Harvnb|Krantz|1991|p=173}}.</ref>

: For all <math>\varepsilon > 0</math> there exists <math>\delta > 0</math> such that, for any tagged partition <math>[a, b]</math> with mesh less than <math>\delta</math>,

: <math>\left| S - \sum_{i=1}^n f(t_i) \, \Delta_i \right| < \varepsilon.</math>

When the chosen tags are the maximum (respectively, minimum) value of the function in each interval, the Riemann sum becomes an upper (respectively, lower) [[Darboux integral|Darboux sum]], suggesting the close connection between the Riemann integral and the [[Darboux integral]].

=== Lebesgue integral ===
{{Main|Lebesgue integration}}
[[File:Lebesgueintegralsimplefunctions finer-dotted.svg|alt=Comparison of Riemann and Lebesgue integrals|thumb|250x250px|Lebesgue integration]]
It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not 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.<ref>{{Harvnb|Rudin|1987|p=5}}.</ref>

Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus [[Henri Lebesgue]] introduced the integral bearing his name, explaining this integral thus in a letter to [[Paul Montel]]:<ref>{{Harvnb|Siegmund-Schultze|2008|p=796}}.</ref>

{{blockquote|I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.|title=|source=}}

As Folland puts it, "To compute the Riemann integral of {{mvar|f}}, one partitions the domain {{closed-closed|''a'', ''b''}} into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of {{mvar|f}} ".<ref>{{Harvnb|Folland|1999|pp=57–58}}.</ref> The definition of the Lebesgue integral thus begins with a [[Measure (mathematics)|measure]], μ. In the simplest case, the [[Lebesgue measure]] {{math|''μ''(''A'')}} of an interval {{math|1=''A'' = [''a'', ''b'']}} is its width, {{math|''b'' − ''a''}}, so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist.<ref>{{Harvnb|Bourbaki|2004|p=IV.43}}.</ref> 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 {{mvar|f}} " philosophy, the integral of a non-negative function {{math|''f'' : '''R''' → '''R'''}} should be the sum over {{mvar|t}} of the areas between a thin horizontal strip between {{math|1=''y'' = ''t''}} and {{math|1=''y'' = ''t'' + ''dt''}}. This area is just {{math|''μ''{ ''x'' : ''f''(''x'') > ''t''} ''dt''}}. Let {{math|1=''f''<sup>∗</sup>(''t'') = ''μ''{ ''x'' : ''f''(''x'') > ''t'' }<nowiki/>}}. The Lebesgue integral of {{mvar|f}} is then defined by

: <math>\int f = \int_0^\infty f^*(t)\,dt</math>

where the integral on the right is an ordinary improper Riemann integral ({{math|''f''{{i sup|∗}}}} is a strictly decreasing positive function, and therefore has a [[well-defined]] improper Riemann integral).<ref>{{Harvnb|Lieb|Loss|2001|p=14}}.</ref> For a suitable class of functions (the [[measurable function]]s) this defines the Lebesgue integral.

A general measurable function {{mvar|f}} is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of {{mvar|f}} and the {{mvar|x}}-axis is finite:<ref>{{Harvnb|Folland|1999|p=53}}.</ref>

: <math>\int_E |f|\,d\mu < + \infty.</math>

In that case, the integral is, as in the Riemannian case, the difference between the area above the {{mvar|x}}-axis and the area below the {{mvar|x}}-axis:<ref name=":3">{{Harvnb|Rudin|1987|p=25}}.</ref>

: <math>\int_E f \,d\mu = \int_E f^+ \,d\mu - \int_E f^- \,d\mu</math>

where

: <math>\begin{alignat}{3}
 & f^+(x) &&{}={} \max \{f(x),0\} &&{}={} \begin{cases}
               f(x), & \text{if } f(x) > 0, \\
               0, & \text{otherwise,}
             \end{cases}\\
 & f^-(x) &&{}={} \max \{-f(x),0\} &&{}={} \begin{cases}
               -f(x), & \text{if } f(x) < 0, \\
               0, & \text{otherwise.}
             \end{cases}
\end{alignat}</math>

=== 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]]. 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 integration|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 integration|Lebesgue–Stieltjes integral]] without depending on [[Measure (mathematics)|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 [[Khinchin integral]], named after [[Aleksandr Khinchin]].
* The [[Itô integral]] and [[Stratonovich integral]], which define integration with respect to [[semimartingale]]s such as [[Wiener process|Brownian motion]].
* The [[Young integral]], which is a kind of Riemann–Stieltjes integral with respect to certain functions of [[Bounded variation|unbounded variation]].
* The [[rough path]] integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both [[semimartingale]]s and processes such as the [[fractional Brownian motion]].
* The [[Choquet integral]], a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953.
* The [[Bochner integral]], a generalization of the Lebesgue integral to functions that take values in a [[Banach space]].

==Properties==

===Linearity===
The collection of Riemann-integrable functions on a closed interval {{math|[''a'', ''b'']}} forms a [[vector space]] under the operations of [[pointwise addition]] and multiplication by a scalar, and the operation of integration

: <math> f \mapsto \int_a^b f(x) \; dx</math>

is a [[linear functional]] on this vector space. Thus, the collection of integrable functions is closed under taking [[linear combination]]s, and the integral of a linear combination is the linear combination of the integrals:<ref name=":0">{{Harvnb|Apostol|1967|p=80}}.</ref>

: <math> \int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \,</math>

Similarly, the set of [[Real number|real]]-valued Lebesgue-integrable functions on a given [[Measure (mathematics)|measure space]] {{mvar|E}} with measure {{mvar|μ}} is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral

: <math> f\mapsto \int_E f \, d\mu </math>

is a linear functional on this vector space, so that:<ref name=":3" />

: <math> \int_E (\alpha f + \beta g) \, d\mu = \alpha \int_E f \, d\mu + \beta \int_E g \, d\mu. </math>

More generally, consider the vector space of all [[measurable function]]s on a measure space {{math|(''E'',''μ'')}}, taking values in a [[Locally compact space|locally compact]] [[Complete metric space|complete]] [[topological vector space]] {{mvar|V}} over a locally compact [[Topological ring|topological field]] {{math|''K'', ''f'' : ''E'' → ''V''}}. Then one may define an abstract integration map assigning to each function {{mvar|f}} an element of {{mvar|V}} or the symbol {{math|''∞''}},

: <math> f\mapsto\int_E f \,d\mu, \,</math>

that is compatible with linear combinations.<ref>{{Harvnb|Rudin|1987|p=54}}.</ref> In this situation, the linearity holds for the subspace of functions whose integral is an element of {{mvar|V}} (i.e. "finite"). The most important special cases arise when {{mvar|K}} is {{math|'''R'''}}, {{math|'''C'''}}, or a finite extension of the field {{math|'''Q'''<sub>''p''</sub>}} of [[p-adic number]]s, and {{mvar|V}} is a finite-dimensional vector space over {{mvar|K}}, and when {{math|''K'' {{=}} '''C'''}} and {{mvar|V}} is a complex [[Hilbert space]].

Linearity, together with some natural continuity properties and normalization for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of [[Daniell integral|Daniell]] for the case of real-valued functions on a set {{mvar|X}}, generalized by [[Nicolas Bourbaki]] to functions with values in a locally compact topological vector space. See {{Harvnb|Hildebrandt|1953}} for an axiomatic characterization of the integral.

=== Inequalities ===
A number of general inequalities hold for Riemann-integrable [[Function (mathematics)|functions]] defined on a [[Closed set|closed]] and [[Bounded set|bounded]] [[Interval (mathematics)|interval]] {{closed-closed|''a'', ''b''}} and can be generalized to other notions of integral (Lebesgue and Daniell).

* ''Upper and lower bounds.'' An integrable function {{mvar|f}} on {{closed-closed|''a'', ''b''}}, is necessarily [[Bounded function|bounded]] on that interval. Thus there are [[real number]]s {{mvar|m}} and {{mvar|M}} so that {{math|''m'' ≤ ''f'' (''x'') ≤ ''M''}} for all {{mvar|x}} in {{closed-closed|''a'', ''b''}}. Since the lower and upper sums of {{mvar|f}} over {{closed-closed|''a'', ''b''}} are therefore bounded by, respectively, {{math|''m''(''b'' − ''a'')}} and {{math|''M''(''b'' − ''a'')}}, it follows that <math display="block"> m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a). </math>
* ''Inequalities between functions.''<ref>{{Harvnb|Apostol|1967|p=81}}.</ref> If {{math|''f''(''x'') ≤ ''g''(''x'')}} for each {{mvar|x}} in {{closed-closed|''a'', ''b''}} then each of the upper and lower sums of {{mvar|f}} is bounded above by the upper and lower sums, respectively, of {{mvar|g}}. Thus <math display="block"> \int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx. </math> This is a generalization of the above inequalities, as {{math|''M''(''b'' − ''a'')}} is the integral of the constant function with value {{mvar|M}} over {{closed-closed|''a'', ''b''}}. In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if {{math|''f''(''x'') < ''g''(''x'')}} for each {{mvar|x}} in {{closed-closed|''a'', ''b''}}, then <math display="block"> \int_a^b f(x) \, dx < \int_a^b g(x) \, dx. </math>
* ''Subintervals.'' If {{closed-closed|''c'', ''d''}} is a subinterval of {{closed-closed|''a'', ''b''}} and {{math|''f''&hairsp;(''x'')}} is non-negative for all {{mvar|x}}, then <math display="block"> \int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx. </math>
* ''Products and absolute values of functions.'' If {{mvar|f}} and {{mvar|g}} are two functions, then we may consider their [[pointwise product]]s and powers, and [[absolute value]]s: <math display="block">
 (fg)(x)= f(x) g(x), \; f^2 (x) = (f(x))^2, \; |f| (x) = |f(x)|.</math> If {{mvar|f}} is Riemann-integrable on {{closed-closed|''a'', ''b''}} then the same is true for {{math|{{abs|''f''}}}}, and <math display="block">\left| \int_a^b f(x) \, dx \right| \leq \int_a^b | f(x) | \, dx. </math> Moreover, if {{mvar|f}} and {{mvar|g}} are both Riemann-integrable then {{math|''fg''}} is also Riemann-integrable, and <math display="block">\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). </math> 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 space|inner product]] of two [[Square-integrable function|square-integrable]] functions {{mvar|f}} and {{mvar|g}} on the interval {{closed-closed|''a'', ''b''}}.
* ''Hölder's inequality''.<ref name=":4">{{Harvnb|Rudin|1987|p=63}}.</ref> Suppose that {{mvar|p}} and {{mvar|q}} are two real numbers, {{math|1 ≤ ''p'', ''q'' ≤ ∞}} with {{math|1={{sfrac|1|''p''}} + {{sfrac|1|''q''}} = 1}}, and {{mvar|f}} and {{mvar|g}} are two Riemann-integrable functions. Then the functions {{math|{{abs|''f''}}<sup>''p''</sup>}} and {{math|{{abs|''g''}}<sup>''q''</sup>}} are also integrable and the following [[Hölder's inequality]] holds: <math display="block">\left|\int f(x)g(x)\,dx\right| \leq
\left(\int \left|f(x)\right|^p\,dx \right)^{1/p} \left(\int\left|g(x)\right|^q\,dx\right)^{1/q}.</math> For {{math|1=''p'' = ''q'' = 2}}, Hölder's inequality becomes the Cauchy–Schwarz inequality.
* ''Minkowski inequality''.<ref name=":4" /> Suppose that {{math|''p'' ≥ 1}} is a real number and {{mvar|f}} and {{mvar|g}} are Riemann-integrable functions. Then {{math|{{abs| ''f'' }}<sup>''p''</sup>, {{abs| ''g'' }}<sup>''p''</sup>}} and {{math|{{abs| ''f'' + ''g'' }}<sup>''p''</sup>}} are also Riemann-integrable and the following [[Minkowski inequality]] holds: <math display="block">\left(\int \left|f(x)+g(x)\right|^p\,dx \right)^{1/p} \leq
\left(\int \left|f(x)\right|^p\,dx \right)^{1/p} +
\left(\int \left|g(x)\right|^p\,dx \right)^{1/p}.</math> An analogue of this inequality for Lebesgue integral is used in construction of [[Lp space|L<sup>p</sup> spaces]].

=== Conventions ===
In this section, {{mvar|f}} is a [[Real number|real-valued]] Riemann-integrable [[Function (mathematics)|function]]. The integral

: <math> \int_a^b f(x) \, dx </math>

over an interval {{math|[''a'', ''b'']}} is defined if {{math|''a'' &lt; ''b''}}. This means that the upper and lower sums of the function {{mvar|f}} are evaluated on a partition {{math|''a'' {{=}} ''x''<sub>0</sub> ≤ ''x''<sub>1</sub> ≤ . . . ≤ ''x''<sub>''n''</sub> {{=}} ''b''}} whose values {{math|''x''<sub>''i''</sub>}} are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating {{mvar|f}} within intervals {{math|[''x''<sub> ''i''</sub> , ''x''<sub> ''i'' +1</sub>]}} where an interval with a higher index lies to the right of one with a lower index. The values {{mvar|a}} and {{mvar|b}}, the end-points of the [[Interval (mathematics)|interval]], are called the [[limits of integration]] of {{mvar|f}}. Integrals can also be defined if {{math|''a'' &gt; ''b''}}:''<ref name=":1" />''

:<math>\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx. </math>

With {{math|''a'' {{=}} ''b''}}, this implies:

:<math>\int_a^a f(x) \, dx = 0. </math>

The first convention is necessary in consideration of taking integrals over subintervals of {{math|[''a'', ''b'']}}; the second says that an integral taken over a degenerate interval, or a [[Point (geometry)|point]], should be [[0 (number)|zero]]. One reason for the first convention is that the integrability of {{mvar|f}} on an interval {{math|[''a'', ''b'']}} implies that {{mvar|f}} is integrable on any subinterval {{math|[''c'', ''d'']}}, but in particular integrals have the property that if {{mvar|c}} is any [[Element (mathematics)|element]] of {{math|[''a'', ''b'']}}, then:''<ref name=":0" />''

:<math> \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.</math>

With the first convention, the resulting relation

: <math>\begin{align}
 \int_a^c f(x) \, dx &{}= \int_a^b f(x) \, dx - \int_c^b f(x) \, dx \\
 &{} = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx
\end{align}</math>

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

== Fundamental theorem of calculus ==

{{Main|Fundamental theorem of calculus}}
The ''fundamental theorem of calculus'' is the statement that [[Derivative|differentiation]] and integration are inverse operations: if a [[continuous function]] is first integrated and then differentiated, the original function is retrieved.<ref>{{Harvnb|Apostol|1967|p=202}}.</ref> 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.<ref>{{Harvnb|Apostol|1967|p=205}}.</ref>

=== First theorem ===
Let {{mvar|f}} be a continuous real-valued function defined on a [[Interval (mathematics)#Definitions|closed interval]] {{math|[''a'', ''b'']}}. Let {{mvar|F}} be the function defined, for all {{mvar|x}} in {{math|[''a'', ''b'']}}, by{{sfn|Montesinos|Zizler|Zizler|2015|p=355}}

: <math>F(x) = \int_a^x f(t)\, dt.</math>

Then, {{mvar|F}} is continuous on {{math|[''a'', ''b'']}}, differentiable on the open interval {{math|(''a'', ''b'')}}, and

: <math>F'(x) = f(x)</math>

for all {{mvar|x}} in {{math|(''a'', ''b'')}}.

=== Second theorem ===
Let {{mvar|f}} be a real-valued function defined on a [[closed interval]] [{{math|''a'', ''b''}}] that admits an [[antiderivative]] {{mvar|F}} on {{math|[''a'', ''b'']}}. That is, {{mvar|f}} and {{mvar|F}} are functions such that for all {{mvar|x}} in {{math|[''a'', ''b'']}},

: <math>f(x) = F'(x).</math>

If {{mvar|f}} is integrable on {{math|[''a'', ''b'']}} then

: <math>\int_a^b f(x)\,dx = F(b) - F(a).</math>

== Extensions ==

=== Improper integrals ===
{{Main|Improper integral}}
[[File:Improper_integral.svg|right|thumb|The [[improper integral]]<math>\int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} = \pi</math>

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 (mathematics)|limit]] of a [[sequence]] of proper [[Riemann integral]]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:<ref>{{Harvnb|Apostol|1967|p=416}}.</ref>

: <math>\int_a^\infty f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx.</math>

If the integrand is only defined or finite on a half-open interval, for instance {{math|<nowiki>(</nowiki>''a'', ''b''<nowiki>]</nowiki>}}, then again a limit may provide a finite result:<ref>{{Harvnb|Apostol|1967|p=418}}.</ref>

: <math>\int_a^b f(x)\,dx = \lim_{\varepsilon \to 0} \int_{a+\epsilon}^{b} f(x)\,dx.</math>

That is, the improper integral is the [[Limit (mathematics)|limit]] of proper integrals as one endpoint of the interval of integration approaches either a specified [[real number]], or {{math|∞}}, or {{math|−∞}}. In more complicated cases, limits are required at both endpoints, or at interior points.

=== Multiple integration ===
{{Main|Multiple integral}}
[[File:Volume_under_surface.png|right|thumb|Double integral computes volume under a surface <math>z=f(x,y)</math>]]
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 that contains its domain.<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=895}}.</ref> For example, a function in two dimensions depends on two real variables, ''x'' and ''y'', and the integral of a function ''f'' over the rectangle ''R'' given as the [[Cartesian product]] of two intervals <math>R=[a,b]\times [c,d]</math> can be written

: <math>\int_R f(x,y)\,dA</math>

where the differential {{math|''dA''}} indicates that integration is taken with respect to area. This [[double integral]] can be defined using [[Riemann sum]]s, and represents the (signed) volume under the graph of {{math|''z'' {{=}} ''f''(''x'',''y'')}} over the domain ''R''.<ref name=":2">{{Harvnb|Anton|Bivens|Davis|2016|p=896}}.</ref> Under suitable conditions (e.g., if ''f'' is continuous), [[Fubini's theorem]] states that this integral can be expressed as an equivalent iterated integral<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=897}}.</ref>

: <math>\int_a^b\left[\int_c^d f(x,y)\,dy\right]\,dx.</math>

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:<ref name=":2" />

: <math>\iint_R f(x,y) \, dA.</math>

Integration over more general domains is possible. The integral of a function ''f'', with respect to volume, over an ''n-''dimensional region ''D'' of <math>\mathbb{R}^n</math> is denoted by symbols such as:

: <math>\int_D f(\mathbf x) d^n\mathbf x \ = \int_D f\,dV.</math>

=== Line integrals and surface integrals ===
{{Main|Line integral|Surface integral}}
[[File:Line-Integral.gif|right|thumb|A line integral sums together elements along a curve.]]
The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces inside higher-dimensional spaces. Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with [[vector field]]s.

A ''line integral'' (sometimes called a ''path integral'') is an integral where the [[Function (mathematics)|function]] to be integrated is evaluated along a [[curve]].<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=980}}.</ref> Various different line integrals are in use. In the case of a closed curve it is also called a ''contour integral''.

The function to be integrated may be a [[scalar field]] or a [[vector field]]. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly [[arc length]] or, for a vector field, the [[Inner product space|scalar product]] of the vector field with a [[Differential (infinitesimal)|differential]] vector in the curve).<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=981}}.</ref> This weighting distinguishes the line integral from simpler integrals defined on [[Interval (mathematics)|intervals]]. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that [[Mechanical work|work]] is equal to [[force]], {{math|'''F'''}}, multiplied by displacement, {{math|'''s'''}}, may be expressed (in terms of vector quantities) as:<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=697}}.</ref>

: <math>W=\mathbf F\cdot\mathbf s.</math>

For an object moving along a path {{mvar|''C''}} in a [[vector field]] {{math|'''F'''}} such as an [[electric field]] or [[gravitational field]], the total work done by the field on the object is obtained by summing up the differential work done in moving from {{math|'''s'''}} to {{math|'''s''' + ''d'''''s'''}}. This gives the line integral<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=991}}.</ref>

: <math>W=\int_C \mathbf F\cdot d\mathbf s.</math>
[[File:Surface_integral_illustration.svg|right|thumb|The definition of surface integral relies on splitting the surface into small surface elements.]]
A ''surface integral'' generalizes double integrals to integration over a [[Surface (mathematics)|surface]] (which may be a curved set in [[space]]); it can be thought of as the [[Multiple integral|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.<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=1014}}.</ref>

For an example of applications of surface integrals, consider a vector field {{math|'''v'''}} on a surface {{math|''S''}}; that is, for each point {{math|''x''}} in {{math|''S''}}, {{math|'''v'''(''x'')}} is a vector. Imagine that a fluid flows through {{math|''S''}}, such that {{math|'''v'''(''x'')}} determines the velocity of the fluid at {{mvar|x}}. The [[flux]] is defined as the quantity of fluid flowing through {{math|''S''}} in unit amount of time. To find the flux, one need to take the [[dot product]] of {{math|'''v'''}} with the unit [[Normal (geometry)|surface normal]] to {{math|''S''}} at each point, which will give a scalar field, which is integrated over the surface:<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=1024}}.</ref>

: <math>\int_S {\mathbf v}\cdot \,d{\mathbf S}.</math>

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]].

=== Contour integrals ===
{{Main|Contour integration}}
In [[complex analysis]], the integrand is a [[complex-valued function]] of a complex variable {{mvar|z}} instead of a real function of a real variable {{mvar|x}}. When a complex function is integrated along a curve <math>\gamma</math> in the complex plane, the integral is denoted as follows

: <math>\int_\gamma f(z)\,dz.</math>

This is known as a [[contour integral]].

=== Integrals of differential forms ===
{{Main|Differential form}}
{{See also|Volume form|Density on a manifold}}
A [[differential form]] is a mathematical concept in the fields of [[multivariable calculus]], [[differential topology]], and [[tensor]]s. Differential forms are organized by degree. For example, a one-form is a weighted sum of the differentials of the coordinates, such as:

: <math>E(x,y,z)\,dx + F(x,y,z)\,dy + G(x,y,z)\, dz</math>

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

: <math>G(x,y,z) \, dx\wedge dy + E(x,y,z) \, dy\wedge dz + F(x,y,z) \, dz\wedge dx.</math>

Here the basic two-forms <math>dx\wedge dy, dz\wedge dx, dy\wedge dz</math> measure oriented areas parallel to the coordinate two-planes. The symbol <math>\wedge</math> denotes the [[wedge product]], which is similar to the [[cross product]] in the sense that the wedge product of two forms representing oriented lengths represents an oriented area. A two-form can be integrated over an oriented surface, and the resulting integral is equivalent to the surface integral giving the flux of <math>E\mathbf i+F\mathbf j+G\mathbf k</math>.

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]] plays the role of the [[gradient]] and [[Curl (mathematics)|curl]] of vector calculus, and [[Generalized Stokes theorem|Stokes' theorem]] simultaneously generalizes the three theorems of vector calculus: the [[divergence theorem]], [[Green's theorem]], and the [[Kelvin-Stokes theorem]].

=== Summations ===
{{Main|Summation#Approximation by definite integrals}}
The discrete equivalent of integration is [[summation]]. Summations and integrals can be put on the same foundations using the theory of [[Lebesgue integral]]s or [[time-scale calculus]].

=== Functional integrals ===
{{Main article|Functional integration}}
An integration that is performed not over a variable (or, in physics, over a space or time dimension), but over a [[Function space|space of functions]], is referred to as a [[functional integral]].

== Applications ==
Integrals are used extensively in many areas. For example, in [[probability theory]], integrals are used to determine the probability of some [[random variable]] falling within a certain range.<ref>{{Harvnb|Feller|1966|p=1}}.</ref> Moreover, the integral under an entire [[probability density function]] must equal 1, which provides a test of whether a [[Function (mathematics)|function]] with no negative values could be a density function or not.<ref>{{Harvnb|Feller|1966|p=3}}.</ref>

Integrals can be used for computing the [[area]] of a two-dimensional region that has a curved boundary, as well as [[Volume integral|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.<ref>{{Harvnb|Apostol|1967|pp=88–89}}.</ref> 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, <math>\pi r^2 h </math>, where <math>r</math> is the radius. In the case of a simple disc created by rotating a curve about the {{Math|''x''}}-axis, the radius is given by {{Math|''f''(''x'')}}, and its height is the differential {{Math|''dx''}}. Using an integral with bounds {{Math|''a''}} and {{Math|''b''}}, the volume of the disc is equal to:<ref>{{Harvnb|Apostol|1967|pp=111–114}}.</ref><math display="block">\pi \int_a^b f^2 (x) \, dx.</math>Integrals are also used in physics, in areas like [[kinematics]] to find quantities like [[Displacement (vector)|displacement]], [[time]], and [[velocity]]. For example, in [[rectilinear motion]], the displacement of an object over the time interval <math>[a,b]</math> is given by
: <math>x(b)-x(a) = \int_a^b v(t) \,dt,</math>

where <math>v(t)</math> is the velocity expressed as a function of time.<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=306}}.</ref> The work done by a force <math>F(x)</math> (given as a function of position) from an initial position <math>A</math> to a final position <math>B</math> is:<ref>{{Harvnb|Apostol|1967|p=116}}.</ref>

: <math>W_{A\rightarrow B} = \int_A^B F(x)\,dx.</math>

Integrals are also used in [[thermodynamics]], 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]]. Let {{math|''f''(''x'')}} be the function of {{mvar|x}} to be integrated over a given interval {{math|[''a'', ''b'']}}. Then, find an antiderivative of {{mvar|f}}; that is, a function {{mvar|F}} such that {{math|''F''′ {{=}} ''f''}} on the interval. Provided the integrand and integral have no [[Mathematical singularity|singularities]] on the path of integration, by the fundamental theorem of calculus,

:<math>\int_a^b f(x)\,dx=F(b)-F(a).</math>

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]], [[integration by parts]], [[Trigonometric substitution|integration by trigonometric substitution]], and [[Partial fractions in integration|integration by partial fractions]].

Alternative methods exist to compute more complex integrals. Many [[nonelementary integral]]s can be expanded in a [[Taylor series]] and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution using [[Meijer G-function]]s 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]].

Computations of volumes of [[solid of revolution|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 [[Lists of integrals|list of integrals]].

===Symbolic===
{{Main|Symbolic integration}}

Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive [[Lists of integrals|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]]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]] and [[Maple (software)|Maple]].

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 expression|closed form]] involving only [[elementary function]]s, include [[rational function|rational]] and [[exponential function|exponential]] functions, [[logarithm]], [[trigonometric functions]] and [[inverse trigonometric functions]], 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 the integral if is elementary. However, functions with closed expressions of antiderivatives are the exception, and consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition and to find the symbolic answer whenever it exists. The Risch algorithm, implemented in [[Mathematica]], [[Maple (software)|Maple]] and other [[computer algebra system]]s, does just that for functions and antiderivatives built from rational functions, [[Nth root|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]] (like the [[Legendre function]]s, the [[hypergeometric function]], the [[gamma function]], the [[incomplete gamma function]] and so on). Extending Risch's algorithm to include such functions is possible but challenging and has been an active research subject.

More recently a new approach has emerged, using [[D-finite function| ''D''-finite functions]], which are the solutions of [[linear differential equation]]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.

Rule-based integration systems facilitate integration.  Rubi, a computer algebra system rule-based integrator, pattern matches an extensive system of symbolic integration rules to integrate a wide variety of integrands.  This system uses over 6600 integration rules to compute integrals.{{sfn|Rich|Scheibe|Abbasi|2018}}  The [[Ramanujan's master theorem#Bracket integration method| method of brackets]] is a generalization of Ramanujan's master theorem that can be applied to a wide range of univariate and multivariate integrals.  A set of rules are applied to the coefficients and exponential terms of the integrand's power series expansion to determine the integral.  The method is closely related to the [[Mellin transform]].{{sfn|Gonzalez|Jiu|Moll|2020}}

===Numerical===
{{Main|Numerical integration}}
[[File:Numerical_quadrature_4up.png|right|thumb|Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method, Gaussian quadrature]]
Definite integrals may be approximated using several methods of [[numerical integration]]. 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]], 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.<ref>{{Harvnb|Dahlquist|Björck|2008|pp=519–520}}.</ref> 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.<ref>{{Harvnb|Dahlquist|Björck|2008|pp=522–524}}.</ref>

Riemann sums, the trapezoidal rule, and Simpson's rule are examples of a family of quadrature rules called the [[Newton–Cotes formulas]]. The degree {{mvar|n}} Newton–Cotes quadrature rule approximates the polynomial on each subinterval by a degree ''{{mvar|n}}'' polynomial. This polynomial is chosen to interpolate the values of the function on the interval.<ref>{{Harvnb|Kahaner|Moler|Nash|1989|p=144}}.</ref> 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]]. One solution to this problem is [[Clenshaw–Curtis quadrature]], in which the integrand is approximated by expanding it in terms of [[Chebyshev polynomials]].

[[Romberg's method]] halves the step widths incrementally, giving trapezoid approximations denoted by {{math|''T''(''h''<sub>0</sub>)}}, {{Math|''T''(''h''<sub>1</sub>)}}, and so on, where {{math|''h''<sub>''k''+1</sub>}} is half of {{math|''h''<sub>''k''</sub>}}. 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 [[Interpolation|interpolate]] a polynomial through the approximations, and extrapolate to {{math|''T''(0)}}. [[Gaussian quadrature]] evaluates the function at the roots of a set of [[orthogonal polynomials]].<ref>{{Harvnb|Kahaner|Moler|Nash|1989|p=147}}.</ref> An {{mvar|n}}-point Gaussian method is exact for polynomials of degree up to {{math|2''n'' − 1}}.

The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives as [[Monte Carlo integration]].<ref>{{Harvnb|Kahaner|Moler|Nash|1989|pp=139–140}}.</ref>

===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 [[displacement (fluid)|displaced]] as the object is submerged.

===Geometrical===
{{main|Quadrature (mathematics)}}
Area can sometimes be found via [[geometrical]] [[compass-and-straightedge construction]]s of an equivalent [[square]].

===Integration by differentiation===
Kempf, Jackson and Morales demonstrated mathematical relations that allow an integral to be calculated by means of [[Derivative|differentiation]]. Their calculus involves the [[Dirac delta function]] and the [[partial derivative]] operator <math>\partial_x</math>. This can also be applied to [[functional integral]]s, allowing them to be computed by [[functional derivative|functional differentiation]].<ref>{{Harvnb|Kempf|Jackson|Morales|2015}}.</ref>

== Examples ==

=== Using the fundamental theorem of calculus ===
The [[fundamental theorem of calculus]] allows straightforward calculations of basic functions:

: <math>\int_0^\pi \sin(x) \,dx = -\cos(x) \big|^{x = \pi}_{x = 0} = -\cos(\pi) - \big(-\cos(0)\big) = 2.</math>

== See also ==
{{Portal|Mathematics}}
* {{annotated link|Integral equation}}
* {{annotated link|Integral symbol}}
* [[Lists of integrals]]

== Notes ==
{{Reflist|group=lower-alpha}}

== References ==
{{Reflist}}

== Bibliography ==
{{refbegin|35em}}
* {{Citation |last1=Anton |first1=Howard |last2=Bivens |first2=Irl C. |last3=Davis |first3=Stephen |title=Calculus: Early Transcendentals |volume= |pages= |year=2016 |edition=11th |publisher=John Wiley & Sons |isbn=978-1-118-88382-2}} 
* {{Citation |last=Apostol |first=Tom M. |title=Calculus, Vol.&nbsp;1: One-Variable Calculus with an Introduction to Linear Algebra |url=https://archive.org/details/calculus01apos |volume= |pages= |year=1967 |edition=2nd |publisher=Wiley |isbn=978-0-471-00005-1 |author-link=Tom M. Apostol}}
* {{Citation |last=Bourbaki |first=Nicolas |title=Integration I |volume= |pages= |year=2004 |publisher=Springer-Verlag |isbn=3-540-41129-1 |author-link=Nicolas Bourbaki}}. In particular chapters III and IV.
* {{Citation |last=Burton |first=David M. |title=The History of Mathematics: An Introduction |volume= |pages= |year=2011 |edition=7th |publisher=McGraw-Hill |isbn=978-0-07-338315-6}}
* {{Citation |last=Cajori |first=Florian |title=A History Of Mathematical Notations Volume II |url=https://archive.org/details/historyofmathema00cajo_0/page/247 |volume= |pages= |year=1929 |publisher=Open Court Publishing |isbn=978-0-486-67766-8 |author-link=Florian Cajori}}
* {{Citation |last1=Dahlquist |first1=Germund |title=Numerical Methods in Scientific Computing, Volume I |volume= |pages= |year=2008 |archive-url=https://web.archive.org/web/20070615185623/http://www.mai.liu.se/~akbjo/NMbook.html |chapter=Chapter&nbsp;5: Numerical Integration |chapter-url=http://www.mai.liu.se/~akbjo/NMbook.html |location=Philadelphia |publisher=[[Society for Industrial and Applied Mathematics|SIAM]] |archive-date=2007-06-15 |last2=Björck |first2=Åke |author1-link=Germund Dahlquist |url-status=dead}}
* {{Citation |last=Feller |first=William |author-link=William Feller |title=An introduction to probability theory and its applications |pages= |year=1966 |publisher=John Wiley & Sons |url=https://archive.org/details/introductiontopr02fell_0 |volume= |url-access=registration}}
* {{Citation |last=Folland |first=Gerald B. |author-link=Gerald Folland |title=Real Analysis: Modern Techniques and Their Applications |volume= |pages= |year=1999 |edition=2nd |publisher=John Wiley & Sons |isbn=0-471-31716-0}}
* {{Citation |last=Fourier |first=Jean Baptiste Joseph |title=Théorie analytique de la chaleur |url=https://books.google.com/books?id=TDQJAAAAIAAJ |volume= |page=§231 |year=1822 |publisher=Chez Firmin Didot, père et fils |author-link=Joseph Fourier}}<br />Available in translation as {{citation |last=Fourier |first=Joseph |title=The analytical theory of heat |url=https://archive.org/details/analyticaltheory00fourrich |volume= |pages=200–201 |year=1878<!--original 1822--> |others=Freeman, Alexander (trans.) |publisher=Cambridge University Press}}
*{{Citation |last1=Gonzalez |first1=Ivan |title=An extension of the method of brackets. Part 2 |date=1 January 2020 |journal=Open Mathematics |volume=18 |issue=1 |pages=983–995 |language=en |doi=10.1515/math-2020-0062 |issn=2391-5455 |last2=Jiu |first2=Lin |last3=Moll |first3=Victor H. |s2cid=222004668 |doi-access=free |arxiv=1707.08942}}
* {{Citation |last= |first= |title=The Works of Archimedes |url=https://archive.org/details/worksofarchimede029517mbp |volume= |pages= |year=2002 |editor-last=Heath |editor-first=T. L. |publisher=Dover |isbn=978-0-486-42084-4 |editor-link=Thomas Little Heath}}<br/>(Originally published by Cambridge University Press, 1897, based on J. L. Heiberg's Greek version.)
* {{Citation |last=Hildebrandt |first=T. H. |title=Integration in abstract spaces |url=http://projecteuclid.org/euclid.bams/1183517761 |journal=[[Bulletin of the American Mathematical Society]] |volume=59 |issue=2 |pages=111–139 |year=1953 |doi=10.1090/S0002-9904-1953-09694-X |issn=0273-0979 |author-link=Theophil Henry Hildebrandt |doi-access=free}}
* {{Citation |last1=Kahaner |first1=David |title=Numerical Methods and Software |url=https://archive.org/details/numericalmethods0000kaha |volume= |pages= |year=1989 |chapter=Chapter&nbsp;5: Numerical Quadrature |publisher=Prentice Hall |isbn=978-0-13-627258-8 |last2=Moler |first2=Cleve |last3=Nash |first3=Stephen |author2-link=Cleve Moler |url-access=registration}}
*{{Citation |last=Kallio |first=Bruce Victor |title=A History of the Definite Integral |url=https://open.library.ubc.ca/soa/cIRcle/collections/ubctheses/831/items/1.0080597 |volume= |pages= |year=1966 |type=M.A. thesis |archive-url=https://web.archive.org/web/20140305054035/https://circle.ubc.ca/bitstream/id/132341/UBC_1966_A8%20K3.pdf |publisher=University of British Columbia |access-date=2014-02-28 |archive-date=2014-03-05 |url-status=dead}}
*{{Citation |last=Katz |first=Victor J. |author-link=Victor J. Katz |title=A History of Mathematics: An Introduction |volume= |pages= |year=2009 |publisher=[[Addison-Wesley]] |isbn=978-0-321-38700-4}}
*{{Citation |last1=Kempf |first1=Achim |last2=Jackson |first2=David M. |last3=Morales |first3=Alejandro H. |title=How to (path-)integrate by differentiating |journal=Journal of Physics: Conference Series |volume=626 |pages=012015 |year=2015 |issue=1 |publisher=[[IOP Publishing]] |doi=10.1088/1742-6596/626/1/012015 |arxiv=1507.04348 |bibcode=2015JPhCS.626a2015K |s2cid=119642596}}
* {{Citation |last=Krantz |first=Steven G. |title=Real Analysis and Foundations |year=1991 |publisher=CRC Press |author-link=Steven G. Krantz |url=https://books.google.com/books?id=OI-0vu1rb7MC&pg=PA173 |isbn=0-8493-7156-2}}
* {{Citation |last=Leibniz |first=Gottfried Wilhelm |title=Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern. Erster Band |url=http://name.umdl.umich.edu/AAX2762.0001.001 |volume= |pages= |year=1899 |editor-last=Gerhardt |editor-first=Karl Immanuel |place=Berlin |publisher=Mayer & Müller |author-link=Gottfried Wilhelm Leibniz}}
* {{citation |last1=Lieb |first1=Elliott |title=Analysis |volume=14 |pages= |year=2001 |series=[[Graduate Studies in Mathematics]] |edition=2nd |publisher=[[American Mathematical Society]] |isbn=978-0821827833 |last2=Loss |first2=Michael |author-link1=Elliott H. Lieb |author2-link=Michael Loss}}
* {{citation |last1=Montesinos |first1=Vicente |last2=Zizler |first2=Peter |last3=Zizler |first3=Václav |title=An Introduction to Modern Analysis |edition=illustrated |publisher=Springer |year=2015 |isbn=978-3-319-12481-0 |url=https://books.google.com/books?id=mlX1CAAAQBAJ&pg=PA355}}
* Paul J. Nahin (2015), ''Inside Interesting Integrals'', Springer, ISBN 978-1-4939-1276-6.
*{{Citation |last1=Rich |first1=Albert |title=Rule-based integration: An extensive system of symbolic integration rules |date=16 December 2018 |journal=Journal of Open Source Software |volume=3 |issue=32 |pages=1073 |doi=10.21105/joss.01073 |last2=Scheibe |first2=Patrick |last3=Abbasi |first3=Nasser |bibcode=2018JOSS....3.1073R |s2cid=56487062 |doi-access=free}}
* {{Citation |last=Rudin |first=Walter |title=Real and Complex Analysis |volume= |pages= |year=1987 |chapter=Chapter&nbsp;1: Abstract Integration |edition=International |publisher=McGraw-Hill |isbn=978-0-07-100276-9 |author-link=Walter Rudin}}
* {{Citation |last=Saks |first=Stanisław |title=Theory of the integral |url=http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez= |volume= |pages= |year=1964 |edition=English translation by L. C. Young. With two additional notes by Stefan Banach. Second revised |place=New York |publisher=Dover |author-link=Stanisław Saks}}
* {{citation |last=Siegmund-Schultze |first=Reinhard |author-link=Reinhard Siegmund-Schultze |title=Princeton Companion to Mathematics |volume= |pages= |year=2008 |editor=Timothy Gowers |chapter=Henri Lebesgue |publisher=Princeton University Press |isbn=978-0-691-11880-2 |editor2=June Barrow-Green |editor3=Imre Leader}}.
* {{Citation |last=Stillwell |first=John |author-link=John Stillwell |title=Mathematics and Its History |year=1989 |publisher=Springer |isbn=0-387-96981-0 |url=https://archive.org/details/mathematicsitshi0000stil |url-access=registration}}
* {{Citation |last1=Stoer |first1=Josef |author-link=Josef Stoer |title=Introduction to Numerical Analysis |volume= |pages= |year=2002 |chapter=Topics in Integration |edition=3rd |publisher=Springer |isbn=978-0-387-95452-3 |last2=Bulirsch |first2=Roland |author-link2=Roland Bulirsch}}.
* {{citation |editor-last=Struik |editor-first=Dirk Jan |last= |first= |title=A Source Book in Mathematics, 1200-1800 |volume= |pages= |year=1986 |publisher=Princeton University Press |place=Princeton, New Jersey |isbn=0-691-08404-1 |editor-link=Dirk Jan Struik}}
* Cornel Ioan Vălean (2019), ''(Almost Impossible) Integrals, Sums, and Series'', Springer, ISBN 978-3-030-02461-1.
* Cornel Ioan Vălean (2023), ''More (Almost Impossible) Integrals, Sums, and Series'', Springer, ISBN 978-3-031-21261-1.
* {{Citation |website=W3C |title=Arabic mathematical notation<!--W3C Interest Group Note 31--> |url=http://www.w3.org/TR/arabic-math/ |volume= |pages= |year=2006<!--January-->}}
{{refend}}

==External links==
{{Wikibooks|Calculus}}
* {{springer|title=Integral|id=p/i051340}}
* [http://www.wolframalpha.com/calculators/integral-calculator/ Online Integral Calculator], [[Wolfram Alpha]].

===Online books===
* Keisler, H. Jerome, [http://www.math.wisc.edu/~keisler/calc.html Elementary Calculus: An Approach Using Infinitesimals], University of Wisconsin
* Stroyan, K. D., [https://web.archive.org/web/20050911104158/http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm A Brief Introduction to Infinitesimal Calculus], University of Iowa
* Mauch, Sean, [https://web.archive.org/web/20060415161115/http://www.its.caltech.edu/~sean/book/unabridged.html ''Sean's Applied Math Book''], CIT, an online textbook that includes a complete introduction to calculus
* Crowell, Benjamin, [http://www.lightandmatter.com/calc/ ''Calculus''], Fullerton College, an online textbook
* Garrett, Paul, [http://www.math.umn.edu/~garrett/calculus/ Notes on First-Year Calculus]
* Hussain, Faraz, [http://www.understandingcalculus.com Understanding Calculus], an online textbook
* Johnson, William Woolsey (1909) [http://babel.hathitrust.org/cgi/pt?id=miun.aam9447.0001.001;view=1up;seq=9 Elementary Treatise on Integral Calculus], link from [[HathiTrust]].
* Kowalk, W. P., [http://einstein.informatik.uni-oldenburg.de/20910.html ''Integration Theory''], University of Oldenburg. A new concept to an old problem. Online textbook
* Sloughter, Dan, [http://math.furman.edu/~dcs/book Difference Equations to Differential Equations], an introduction to calculus
* [http://numericalmethods.eng.usf.edu/topics/integration.html Numerical Methods of Integration] at ''Holistic Numerical Methods Institute''
* P. S. Wang, [https://web.archive.org/web/20060917023831/http://www.lcs.mit.edu/publications/specpub.php?id=660 Evaluation of Definite Integrals by Symbolic Manipulation] (1972)&nbsp;— a cookbook of definite integral techniques

{{Integral}}
{{Lp spaces}}
{{Analysis-footer}}
{{Authority control}}
{{Machine learning evaluation metrics}}

[[Category:Integrals| ]]
[[Category:Functions and mappings]]
[[Category:Linear operators in calculus]]