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 \( {\displaystyle D\subset \mathbb {R} ^{3}} \) of a function f(x,y,z), and is usually written as:

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

A volume integral in cylindrical coordinates is

\( {\displaystyle \iiint _{D}f(\rho ,\varphi ,z)\rho \,d\rho \,d\varphi \,dz,} \)

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

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

Example 1

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

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

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 density function on the unit cube then the volume integral will give the total mass of the cube. For example for density function:

\( {\displaystyle {\begin{cases}f:\mathbb {R} ^{3}\to \mathbb {R} \\(x,y,z)\longmapsto x+y+z\end{cases}}} \)

the total mass of the cube is:

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