Volume integral
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In mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3-dimensional domain.
Volume integral is a triple integral of the constant function 1, which gives the volume of the region D, that is, the integral
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:
A volume integral in cylindrical coordinates is
and a volume integral in spherical coordinates has the form
[edit] Example
Integrating the function f(x,y,z) = 1 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 function 
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:
