Volume of a hypersphere

In this appendix we will compute the volume of an -dimensional hypersphere of radius , which we call . This is defined as:

In order to calculate it we will use a "trick". First, we change variable defining so that and thus[1]:
We then have:
Multiplying both sides by and integrating over from to infinity:
The right hand side is equal to:
If we now define , in the integral in the left hand side of the equation we recognise the definition of Euler's function[2]:
This way the volume of an -dimensional sphere of unit radius is , and thanks to the fact that , in the end we have:
Let us verify that this formula returns the values we expect for and . We first note that:
This way:

which is exactly what we expected.
  1. We also use the fact that , since .
  2. As a remainder, the function is defined as:
    If , then .