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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.
Next: Gradient
Up: Vector Algebra and Vector
Previous: Vector Line Integrals
Richard Fitzpatrick
2011-03-31