Volume integral

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

Often the volume integral is represented in terms of a differential volume element <math> dV=dx\, dy\, dz </math>.

<math display="block">\iiint_D f(x,y,z)\,dV.</math>

It can also mean a triple integral within a region <math>D \subset \R^3</math> of a function <math>f(x,y,z),</math> and is usually written as:

<math display="block">\iiint_D f(x,y,z)\,dx\,dy\,dz.</math>

A volume integral in cylindrical coordinates is

<math display="block">\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>\varphi</math> as the azimuth and <math>\theta</math> measured from the polar axis (see more on conventions)) has the form

<math display="block">\iiint_D f(r,\theta,\varphi) r^2 \sin\theta \,dr \,d\theta\, d\varphi .</math>

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 <math> (x,y,z)\mapsto(u,v,w) </math>. We can represent the integral as the following.

<math display="block">\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</math>

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}

</math>

Example

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

<math display="block">\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 display="block"> \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 display="block">\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>

Volume integral

In coordinates

Example

See also

External links