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.

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 (using the convention for angles with \theta as the azimuth and \phi measured from the polar axis (see more on conventions)) has the form

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

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