# 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.

## In coordinates

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

${\displaystyle \iiint \limits _{D}f(x,y,z)\,dx\,dy\,dz.}$

A volume integral in cylindrical coordinates is

${\displaystyle \iiint \limits _{D}f(r,\theta ,z)\,r\,dr\,d\theta \,dz,}$

and a volume integral in spherical coordinates (using the convention for angles with ${\displaystyle \theta }$ as the azimuth and ${\displaystyle \phi }$ measured from the polar axis (see more on conventions)) has the form

${\displaystyle \iiint \limits _{D}f(\rho ,\theta ,\phi )\,\rho ^{2}\sin \phi \,d\rho \,d\theta \,d\phi .}$

## Example 1

Integrating the function ${\displaystyle f(x,y,z)=1}$ over a unit cube yields the following result:

So the volume of the unit cube is 1 as expected. This is rather trivial however, and a volume integral is far more powerful. For instance if we have a scalar function {\displaystyle {\begin{aligned}f\colon \mathbb {R} ^{3}&\to \mathbb {R} \end{aligned}}} describing the density of the cube at a given point ${\displaystyle (x,y,z)}$ by ${\displaystyle f=x+y+z}$ then performing the volume integral will give the total mass of the cube: