Volume integral
Part of a series of articles about  
Calculus  



Specialized 

In mathematics—in particular, in multivariable calculus—a volume integral refers to an integral over a 3dimensional 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 D in R^{3} 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
Example 1
Integrating the function 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 by then performing the volume integral will give the total mass of the cube:
See also
External links
 Hazewinkel, Michiel, ed. (2001), "Multiple integral", Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Weisstein, Eric W. "Volume integral". MathWorld.