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Volume integral

Integral over a 3-D domain

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated May 12, 2025

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.

01In coordinates

Often the volume integral is represented in terms of a differential volume element dV=dx\,dy\,dz. \iiint _{D}f(x,y,z)\,dV. 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 . The triple integral can be transformed from Cartesian coordinates to any arbitrary coordinate system using the Jacobian matrix and determinant. Suppose we have a transformation of coordinates from (x,y,z)\mapsto (u,v,w). We can represent the integral as the following. \iiint _{D}f(x,y,z)\,dx\,dy\,dz=\iiint _{D}f(u,v,w)\left|{\frac {\partial (x,y,z)}{\partial (u,v,w)}}\right|\,du\,dv\,dw Where we define the Jacobian determinant to be. \mathbf {J} ={\frac {\partial (x,y,z)}{\partial (u,v,w)}}={\begin{vmatrix}{\frac {\partial x}{\partial u}}&{\frac {\partial x}{\partial v}}&{\frac {\partial x}{\partial w}}\\{\frac {\partial y}{\partial u}}&{\frac {\partial y}{\partial v}}&{\frac {\partial y}{\partial w}}\\{\frac {\partial z}{\partial u}}&{\frac {\partial z}{\partial v}}&{\frac {\partial z}{\partial w}}\\\end{vmatrix}}

02Example

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}}

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Sources and credits

This article is adapted from the Wikipedia article “Volume integral”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed. More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region enclosed by the surface.

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Volume element

In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form d V = ρ d u 1 d u 2 d u 3 {\displaystyle \mathrm {d} V=\rho (u_{1},u_{2},u_{3})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}} where the u i {\displaystyle u_{i}} are the coordinates, so that the volume of any set B {\displaystyle B} can be computed by Volume ⁡ ( B ) = ∫ B ρ ( u 1 , u 2 , u 3 ) d u 1 d u 2 d u 3 .