Volume integral

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In mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3-dimensional domain.

Volume integral is a triple integral of the constant function 1, which gives the volume of the region D, that is, the integral

\operatorname{Vol}(D)=\iiint\limits_D dx\,dy\,dz.

It can also mean a triple integral within a region D in R3 of a function f(x,y,z), and is usually written as:

\iiint\limits_D f(x,y,z)\,dx\,dy\,dz.

A volume integral in cylindrical coordinates is

\iiint\limits_D f(r,\theta,z)\,r\,dr\,d\theta\,dz,

and a volume integral in spherical coordinates has the form

\iiint\limits_D f(\rho,\theta,\phi)\,\rho^2 \sin\phi \,d\rho \,d\phi\, d\theta .

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