Volume integral

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 .[1][2][3]

In coordinatesEdit

In Cartesian coordinates, a volume integral can be expressed as a triple integral within a region   of a function   and is usually written as:[4]

 

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).

ExampleEdit

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:

 

See alsoEdit

Integral as volumeEdit

ReferencesEdit

  1. ^ "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-09-19.
  2. ^ "Calculus III - Triple Integrals". tutorial.math.lamar.edu. Retrieved 2020-09-19.
  3. ^ "Triple integrals (article)". Khan Academy. Retrieved 2020-09-19.
  4. ^ Weisstein, Eric W. "Volume Integral". mathworld.wolfram.com. Retrieved 2020-09-19.

External linksEdit