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.
It can also mean a triple integral within a region
and is usually written as:
A volume integral in cylindrical coordinates is
f ( ρ , φ , z ) ρ
d φ
{\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
φ
θ
measured from the polar axis (see more on conventions)) has the form
f ( r , θ , φ )
sin θ
d θ
d φ .
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: