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A volume integral takes the form

(1341) 
where is some volume, and
is a small volume element. The
volume element is sometimes written , or even .
As an example
of a volume integral, let us evaluate the center of gravity of a solid pyramid. Suppose that
the pyramid has a square base of side , a height , and is composed of material of uniform density. Let the centroid of the base lie at the origin, and let
the apex lie at .
By symmetry, the center of mass lies on the line joining the centroid to the apex.
In fact, the height of the center of mass is given by

(1342) 
The bottom integral is just the volume of the pyramid, and can be written
Here, we have evaluated the integral last because the limits of the  and  integrals are dependent.
The top integral takes the form
Thus,

(1345) 
In other words, the center of mass of a pyramid lies one quarter of the way between the centroid of the base and the apex.
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Up: Vector Algebra and Vector
Previous: Vector Line Integrals
Richard Fitzpatrick
20110331