Volume integral

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In mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3-dimensional domain.

Volume integral is a triple integral of the constant function 1, which gives the volume of the region D, that is, the integral

[\operatorname(D)=\iiint\limits_D dx\,dy\,dz.]

It can also mean a triple integral within a region D in R³ of a function [f(x,y,z),] and is usually written as:

[\iiint\limits_D f(x,y,z)\,dx\,dy\,dz.]

Parametric Form

[\int\limits_D g\,dV = \iiint\limits_D g(\mathbf(s,t,u)) \, \over \partial s} \cdot \left ( \over \partial t} \times \over \partial u} \right ) \,ds\,dt\,du.]

within which note that

[ = \over \partial s} \cdot \left ( \over \partial t} \times \over \partial u} \right ).]

See also

• divergence theorem
• surface integral
• Volume and surface elements in different co-ordinate systems

External link

• [MathWorld article on volume integrals]

 

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