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
*r^N (pi^(N/2))/((N/2)!)*.
The only hard part is taking the factorial
of a half-integer. The real definition is that *x! = Gamma (x + 1)*, but
if you want a formula, it's:

*(1/2 + n)! = sqrt(pi) ((2n + 2)!)/((n + 1)!4^(n + 1))*

To get the surface area, you just differentiate to get
*N (pi^(N/2))/((N/2)!)r^(N - 1)*.

There is a clever way to obtain this formula using Gaussian
integrals. First, we note that the integral over the line of
*e^(-x^2)* is *sqrt(pi)*. Therefore the integral over *N*-space of
*e^(-x_1^2 - x_2^2 - ... - x_N^2)* is *sqrt(pi)^n*. Now we change to
spherical coordinates. We get the integral from 0 to infinity
of *Vr^(N - 1)e^(-r^2)*, 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''.

Fri Feb 20 21:45:30 EST 1998