Volume integral

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 :$operatorname\{Vol\}(D)=iiintlimits\_D\; dx,dy,dz.$

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:

:$iiintlimits\_D\; f(x,y,z),dx,dy,dz.$

A volume integral in cylindrical coordinates is

:$iiintlimits\_D\; f(r,\; heta,z),r,dr,d\; heta,dz,$

and a volume integral in spherical coordinates has the form

:$iiintlimits\_D\; f(\; ho,\; heta,phi),\; ho^2\; sinphi\; ,d\; ho\; ,dphi,\; d\; heta\; .$

Example

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

$iiint\; limits\_0^1\; 1\; ,dx,\; dy\; ,dz\; =\; iint\; limits\_0^1\; (1\; -\; 0)\; ,dy\; ,dz\; =\; int\; limits\_0^1\; (1\; -\; 0)\; dz\; =\; 1\; -\; 0\; =\; 1$

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 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:

$iiint\; limits\_0^1\; x\; +\; y\; +\; z\; ,\; dx\; ,dy\; ,dz\; =\; iint\; limits\_0^1\; frac\; 12\; +\; y\; +\; z\; ,\; dy\; ,dz\; =\; int\; limits\_0^1\; 1\; +\; z\; ,\; dz\; =\; frac\; 32$

ee also

*divergence theorem
*surface integral
*Volume and surface elements in different co-ordinate systems

External links

* [http://mathworld.wolfram.com/VolumeIntegral.html MathWorld article on volume integrals]