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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.

Contents

In coordinates

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:

∭ D f ( x , y , z ) d x d y d z .

A volume integral in cylindrical coordinates is

∭ D f ( ρ , φ , z ) ρ d ρ d φ d z ,

and a volume integral in spherical coordinates (using the ISO convention for angles with φ as the azimuth and θ measured from the polar axis (see more on conventions)) has the form

∭ D f ( r , θ , φ ) r 2 sin ⁡ φ d r d θ d φ .

Example 1

Integrating the function f ( x , y , z ) = 1 over a unit cube yields the following result:

∫ 0 1 ∫ 0 1 ∫ 0 1 1 d x d y d z = ∫ 0 1 ∫ 0 1 ( 1 − 0 ) d y d z = ∫ 0 1 ( 1 − 0 ) d z = 1 − 0 = 1

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 f : R 3 → R 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:

∫ 0 1 ∫ 0 1 ∫ 0 1 ( x + y + z ) d x d y d z = ∫ 0 1 ∫ 0 1 ( 1 2 + y + z ) d y d z = ∫ 0 1 ( 1 + z ) d z = 3 2

References

Volume integral Wikipedia
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