Volume integral

In mathematics—in particular, in multivariable calculus—a volume integral refers to an integral over a 3-dimensional domain, that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities.

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(\rho ,\varphi ,z)\,\rho \,d\rho \,d\varphi \,dz,

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

\iiint \limits _{D}f(r,\theta ,\varphi )\,r^{2}\sin \varphi \,dr\,d\theta \,d\varphi .

Example 1

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

$\int \limits _{0}^{1}\int \limits _{0}^{1}\int \limits _{0}^{1}1\,dx\,dy\,dz=\int \limits _{0}^{1}\int \limits _{0}^{1}(1-0)\,dy\,dz=\int \limits _{0}^{1}(1-0)dz=1-0=1$

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:

$\int \limits _{0}^{1}\int \limits _{0}^{1}\int \limits _{0}^{1}\left(x+y+z\right)\,dx\,dy\,dz=\int \limits _{0}^{1}\int \limits _{0}^{1}\left({\frac {1}{2}}+y+z\right)\,dy\,dz=\int \limits _{0}^{1}\left(1+z\right)\,dz={\frac {3}{2}}$