Volume integral explained

In mathematics (particularly multivariable calculus), a volume integral (∭) is 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, or to calculate mass from a corresponding density function.

In coordinates

It can also mean a triple integral within a region

D \subset \R ^{3} of a function f(x,y,z), and is usually written as:$$\backslash iiint\_D\; f(x,y,z)\backslash ,dx\backslash ,dy\backslash ,dz.$$

A volume integral in cylindrical coordinates is$$\backslash iiint\_D\; f(\backslash rho,\backslash varphi,z)\; \backslash rho\; \backslash ,d\backslash rho\; \backslash ,d\backslash varphi\; \backslash ,dz,$$and a volume integral in spherical coordinates (using the ISO convention for angles with

\varphi as the azimuth and \theta measured from the polar axis (see more on conventions)) has the form$$\backslash iiint\_D\; f(r,\backslash theta,\backslash varphi)\; r^2\; \backslash sin\backslash theta\; \backslash ,dr\; \backslash ,d\backslash theta\backslash ,\; d\backslash varphi\; .$$

Example

Integrating the equation

f(x,y,z) = 1 over a unit cube yields the following result:$$\backslash int\_0^1\; \backslash int\_0^1\; \backslash int\_0^1\; 1\; \backslash ,dx\; \backslash ,dy\; \backslash ,dz\; =\; \backslash int\_0^1\; \backslash int\_0^1\; (1\; -\; 0)\; \backslash ,dy\; \backslash ,dz\; =\; \backslash int\_0^1\; \backslash left(1\; -\; 0\backslash right)\; dz\; =\; 1\; -\; 0\; =\; 1$$

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 density function on the unit cube then the volume integral will give the total mass of the cube. For example for density function:$$\backslash beginf:\; \backslash R^3\; \backslash to\; \backslash R\; \backslash \backslash f:\; (x,y,z)\; \backslash mapsto\; x+y+z\backslash end$$ the total mass of the cube is: $$\backslash int\_0^1\; \backslash int\_0^1\; \backslash int\_0^1\; (x+y+z)\; \backslash ,dx\; \backslash ,dy\; \backslash ,dz\; =\; \backslash int\_0^1\; \backslash int\_0^1\; \backslash left(\backslash frac\; 1\; 2\; +\; y\; +\; z\backslash right)\; dy\; \backslash ,dz\; =\; \backslash int\_0^1\; (1\; +\; z)\; \backslash ,\; dz\; =\; \backslash frac\; 3\; 2$$

See also

• Divergence theorem
• Surface integral
• Volume element
• Line element
• Line integral