» Required Reading » Table of Contents » Chapter 20 20.10 Volume Integrals
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The idea of splitting a path or line or area or surface into small pieces can be applied just as well to a volume. We therefore define volume integrals by the same idea which by this time, I hope does not need repeating.

Volumes are numbers rather than vectors in 3 dimensions, so the definition is quite straightforward. With integrand 1 the integral becomes the volume itself. Volume integrals with density of whatever as integrand give the total amount of whatever is in the volume integrated over. These are commonly encountered, in particular the volume integral of charge or mass density gives the charge or mass of in that volume.
We also encounter moments. The integral of r² times the mass density over a volume gives its moment of inertia about the z axis. Here r² is the square of the distance of a point to that axis or x²+y². This is a "moment" about an axis.
Such integrals are usually denoted as

Having defined all these entities we now turn to the question: how do we evaluate them?

The basic answer is: we use the fundamental theorem of calculus, which we will now describe, to either give us a direct answer or to reduce the integral in question to our performing one or a sequence of anti-derivatives.

Obviously the definitions here can be extended to higher dimensions.
In every case we can prove by the same argument that integrals of continuous functions over bounded closed regions always exist.