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.

Volume integral is a triple integral of the constant function 1, which gives the volume of the region D, that is, the integral

It can also mean a triple integral within a region D in R³ of a function f(x,y,z), and is usually written as:

A volume integral in cylindrical coordinates is

and a volume integral in spherical coordinates (using the standard convention for angles) has the form

[edit] Example

Integrating the function f(x,y,z) = 1 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 (x,y,z) by f = x + y + z then performing the volume integral will give the total mass of the cube:

[edit] See also

• Divergence theorem
• Surface integral

[edit] External links

• MathWorld article on volume integrals

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