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In mathematics—in particular, in multivariable calculus—a volume integral refers to 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.

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In coordinatesEdit

It can also mean a triple integral within a region D in R3 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 1Edit

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 alsoEdit

External linksEdit