Editing Integral (section)

=== 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>