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In mathematics (particularly
multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather t ...
), a volume integral (∭) refers to an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
for many applications, for example, to calculate
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
densities.


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: \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
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)) 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 f: \R^3 \to \R \\ f: (x,y,z) \mapsto x+y+z \end 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


See also

* Divergence theorem *
Surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, on ...
* Volume element


External links

* * {{Calculus topics Multivariable calculus