Inmathematics (particularlymultivariable calculus), avolume integral (∭) is anintegral over a3-dimensional domain; that is, it is a special case ofmultiple integrals. Volume integrals are especially important inphysics for many applications, for example, to calculateflux densities, or to calculate mass from a corresponding density function.

Often the volume integral is represented in terms of a differential volume element${\displaystyle dV=dxdydz}$.$${\displaystyle {\iiint }_{D}f(x,y,z)dV.}$$It can also mean atriple integral within a region${\displaystyle D\subset {\mathbb{R}}^3}$ of afunction${\displaystyle f(x,y,z),}$ and is usually written as:$${\displaystyle {\iiint }_{D}f(x,y,z)dxdydz.}$$A volume integral incylindrical coordinates is$${\displaystyle {\iiint }_{D}f(\rho ,\phi ,z)\rho d\rho d\phi dz,}$$and a volume integral inspherical coordinates (using the ISO convention for angles with${\displaystyle \phi }$ as the azimuth and${\displaystyle \theta }$ measured from the polar axis (see more onconventions)) has the form$${\displaystyle {\iiint }_{D}f(r,\theta ,\phi )r^2\sin \theta drd\theta d\phi .}$$The triple integral can be transformed from Cartesian coordinates to any arbitrary coordinate system using theJacobian matrix and determinant. Suppose we have a transformation of coordinates from${\displaystyle (x,y,z)\mapsto (u,v,w)}$. We can represent the integral as the following.$${\displaystyle {\iiint }_{D}f(x,y,z)dxdydz={\iiint }_{D}f(u,v,w)\left|\frac{\mathrm{\partial }(x,y,z)}{\mathrm{\partial }(u,v,w)}\right|dudvdw}$$Where we define the Jacobian determinant to be.

Integrating the equation${\displaystyle f(x,y,z)=1}$ over a unit cube yields the following result:$${\displaystyle {\int }_0^1{\int }_0^1{\int }_0^{1}1dxdydz={\int }_0^1{\int }_0^1(1-0)dydz={\int }_0^1\left(1-0\right)dz=1-0=1}$$

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:$${\displaystyle \{\begin{array}{l}f:{\mathbb{R}}^3\to \mathbb{R}\\ f:(x,y,z)\mapsto x+y+z\end{array}}$$ the total mass of the cube is:$${\displaystyle {\int }_0^1{\int }_0^1{\int }_0^1(x+y+z)dxdydz={\int }_0^1{\int }_0^1\left(\frac{1}{2}+y+z\right)dydz={\int }_0^1(1+z)dz=\frac{3}{2}}$$

• Divergence theorem
• Surface integral
• Volume element
• Line element
• Line integral

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

• Multivariable calculus

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