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A volume integral takes the form

(50) 
where is a threedimensional mathematical function, some volume in space, and
an element of this volume. The
volume element is sometimes written .
As an example
of a volume integral, let us evaluate the centre of gravity of a solid hemisphere
of radius (centered on the origin).
The height of the centre of gravity is given by

(51) 
The bottom integral is simply the volume of the hemisphere, which is .
The top integral is most easily evaluated in spherical polar coordinates (, , ), for which
and
. Thus,
giving

(53) 
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Richard Fitzpatrick
20070714