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
We then have:
Multiplying both sides by
and integrating over
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
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
We first note that:
which is exactly what we expected.
- ↑ We also use the fact that , since .
- ↑ As a remainder, the function is defined as:
If , then .