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   of a function   and is usually written as:


A volume integral in cylindrical coordinates is

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



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

the total mass of the cube is:

See also

  • "Multiple integral", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • Weisstein, Eric W. "Volume integral". MathWorld.