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A volume integral takes the form
![\begin{displaymath}
\int\!\int\!\int_V f(x,y,z) dV,
\end{displaymath}](img285.png) |
(89) |
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 centre of gravity of a solid hemisphere
of radius
(centered on the origin).
The height of the centre of gravity is given by
![\begin{displaymath}
\overline{z} = \left. \int\!\int\!\int z dV\right/ \int\!\int\!\int dV.
\end{displaymath}](img290.png) |
(90) |
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
![\begin{displaymath}
\overline{z} = \frac{ \pi a^4}{4}\frac{3}{2\pi a^3}= \frac{3 a}{8}.
\end{displaymath}](img297.png) |
(92) |
Next: Gradient
Up: Vectors
Previous: Vector surface integrals
Richard Fitzpatrick
2006-02-02