Volume integral

In mathematics (particularly multivariable calculus), a volume integral (∭) is 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, or to calculate mass from a corresponding density function.

In coordinates
It can also mean a triple integral within a region $$D \subset \R^3$$ of a function $$f(x,y,z),$$ and is usually written as: $$\iiint_D f(x,y,z)\,dx\,dy\,dz.$$

A volume integral in cylindrical coordinates is $$\iiint_D f(\rho,\varphi,z) \rho \,d\rho \,d\varphi \,dz,$$ 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 $$\iiint_D f(r,\theta,\varphi) r^2 \sin\theta \,dr \,d\theta\, d\varphi .$$

Example
Integrating the equation $$ f(x,y,z) = 1 $$ over a unit cube yields the following result: $$\int_0^1 \int_0^1 \int_0^1 1 \,dx \,dy \,dz = \int_0^1 \int_0^1 (1 - 0) \,dy \,dz = \int_0^1 \left(1 - 0\right) dz = 1 - 0 = 1$$

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: $$ \begin{cases} f: \R^3 \to \R \\ f: (x,y,z) \mapsto x+y+z \end{cases}$$ the total mass of the cube is: $$\int_0^1 \int_0^1 \int_0^1 (x+y+z) \,dx \,dy \,dz = \int_0^1 \int_0^1 \left(\frac 1 2 + y + z\right) dy \,dz = \int_0^1 (1 + z) \, dz = \frac 3 2$$