(A.98) |

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

(A.99) |

The bottom integral is just the volume of the pyramid, and can be written

(A.100) |

Here, we have evaluated the -integral last because the limits of the - and - integrals are -dependent. The top integral takes the form

(A.101) |

Thus,

(A.102) |

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.