In mathematics (particularly 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. The volume integral of a function f over a domain D can be denoted abstractly as .
A volume integral in cylindrical coordinates is
and a volume integral in spherical coordinates has the form
using the ISO convention for angles with as the azimuth, and measured from the polar axis (see also Spherical coordinate system § Conventions for more).
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 result, and a volume integral is in actuality 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 the density function:
the total mass of the cube is:
Integral as volumeEdit
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