This is equivalent to the volume of the *N*-1
solid which comprises the boundary of an *N*-Sphere.

The volume of a ball is the easiest formula to remember: It's . The only hard part is taking the factorial of a half-integer. The real definition is that , but if you want a formula, it's:

To get the surface area, you just differentiate to get .

There is a clever way to obtain this formula using Gaussian
integrals. First, we note that the integral over the line of
is . Therefore the integral over *N*-space of
is . Now we change to
spherical coordinates. We get the integral from 0 to infinity
of , where *V* is the surface volume of a sphere.
Integrate by parts repeatedly to get the desired formula.

It is possible to derive the volume of the sphere from ``first principles''.

Mon Feb 23 16:26:48 EST 1998