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

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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  

Example

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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

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