- In coordinates
- Example 1
- See also
- External links
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
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
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
{{Portal|Mathematics}}- Divergence theorem
- Surface integral
- Volume element
External links
- {{springer|title=Multiple integral|id=p/m065370}}
- {{MathWorld|VolumeIntegral|Volume integral}}
- Paul Septimiu Muntean
- Paul, Serbian Archbishop
- Paul, Serbian Patriarch
- Paul Sewald
- Pauls (given name)
- Paulsgrave Williams
- Paul Shah
- Paul Shalala
- Paul Shanahan (hurler)
- Paul Shapiro (activist)
- Paul & Shark
- Paul Sharp
- Paul Sharp (disambiguation)
- Paul Shaw (design historian)
- Paul Sheard
- Paul Sherlock
- Paul Shewan
- Paul Shinichi Itonaga
- Paul Shouvalov
- Paul Sidwell
- Paul Siebels
- Paul Sieber
- Paul Sievert
- Paul Sieving
- Paul Sigler