# Volume integral

{{#invoke: Sidebar | collapsible }}

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 $f(x,y,z),$ 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

$\iiint \limits _{D}f(\rho ,\theta ,\phi )\,\rho ^{2}\sin \phi \,d\rho \,d\theta \,d\phi .$ ## Example 1

Integrating the function $f(x,y,z)=1$ 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 {\begin{aligned}f\colon {\mathbb {R} }^{3}&\to {\mathbb {R} }\end{aligned}} describing the density of the cube at a given point $(x,y,z)$ by $f=x+y+z$ then performing the volume integral will give the total mass of the cube: