Let be the volume of the sphere centered at
and radius
in
. Prove that for
it holds that
Solution
The volume of the sphere in is given by:
Parametrize the sphere by
taking
It then follows from the Change of Variables formula that the rectangular volume element can be written in spherical coordinates as
Thus,
Hence,
(1)
In particular since we get that:
(2)
Using as well as Wallis’ integral we are able to prove the result. Let us assume that
is even, then:
If is odd we work similarly.