Volume integral
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In mathematics (particularly 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, or to calculate mass from a corresponding density function.
In coordinates
It can also mean a triple integral within a region [math]\displaystyle{ D \subset \R^3 }[/math] of a function [math]\displaystyle{ f(x,y,z), }[/math] and is usually written as: [math]\displaystyle{ \iiint_D f(x,y,z)\,dx\,dy\,dz. }[/math]
A volume integral in cylindrical coordinates is [math]\displaystyle{ \iiint_D f(\rho,\varphi,z) \rho \,d\rho \,d\varphi \,dz, }[/math] and a volume integral in spherical coordinates (using the ISO convention for angles with [math]\displaystyle{ \varphi }[/math] as the azimuth and [math]\displaystyle{ \theta }[/math] measured from the polar axis (see more on conventions)) has the form [math]\displaystyle{ \iiint_D f(r,\theta,\varphi) r^2 \sin\theta \,dr \,d\theta\, d\varphi . }[/math]
Example
Integrating the equation [math]\displaystyle{ f(x,y,z) = 1 }[/math] over a unit cube yields the following result: [math]\displaystyle{ \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 }[/math]
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: [math]\displaystyle{ \begin{cases} f: \R^3 \to \R \\ f: (x,y,z) \mapsto x+y+z \end{cases} }[/math] the total mass of the cube is: [math]\displaystyle{ \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 }[/math]
See also
External links
- Hazewinkel, Michiel, ed. (2001), "Multiple integral", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/m065370
- Weisstein, Eric W.. "Volume integral". http://mathworld.wolfram.com/VolumeIntegral.html.
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