Volume integral

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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[edit]

It can also mean a triple integral within a region ${\displaystyle D\subset \mathbb {R} ^{3}}$ of a function ${\displaystyle f(x,y,z),}$ and is usually written as:

A volume integral in cylindrical coordinates is

and a volume integral in spherical coordinates (using the ISO convention for angles with ${\displaystyle \varphi }$ as the azimuth and ${\displaystyle \theta }$ measured from the polar axis (see more on conventions)) has the form

Example[edit]

Integrating the equation ${\displaystyle f(x,y,z)=1}$ over a unit cube yields the following result:

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:

the total mass of the cube is:

See also[edit]

• Divergence theorem
• Surface integral
• Volume element

External links[edit]

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