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, 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[edit]

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 1[edit]

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[edit]

External links[edit]

  • Hazewinkel, Michiel, ed. (2001) [1994], "Multiple integral", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • Weisstein, Eric W. "Volume integral". MathWorld.