Editing Integral (section)

== 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]].