Volume integral

Differential

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Integral

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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 $D\subset \mathbb {R} ^{3}$ of a function $f(x,y,z),$ and is usually written as:

\iiint _{D}f(x,y,z)\,dx\,dy\,dz.

A volume integral in cylindrical coordinates is

\iiint _{D}f(\rho ,\varphi ,z)\rho \,d\rho \,d\varphi \,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

\iiint _{D}f(r,\theta ,\varphi )r^{2}\sin \theta \,dr\,d\theta \,d\varphi .

Example

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

\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}1\,dx\,dy\,dz=\int _{0}^{1}\int _{0}^{1}(1-0)\,dy\,dz=\int _{0}^{1}\left(1-0\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:

{\begin{cases}f:\mathbb {R} ^{3}\to \mathbb {R} \\f:(x,y,z)\mapsto x+y+z\end{cases}}

the total mass of the cube is:

\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}(x+y+z)\,dx\,dy\,dz=\int _{0}^{1}\int _{0}^{1}\left({\frac {1}{2}}+y+z\right)dy\,dz=\int _{0}^{1}(1+z)\,dz={\frac {3}{2}}

External links

"Multiple integral", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Weisstein, Eric W. "Volume integral". MathWorld.

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Volume integral

In coordinates

Example

See also

External links