Problem 16P

Derive a formula for the volume of a d-dimensional hypersphere.

Solution 16P

The Gaussian function has an antiderivative and if it is integrated from to , its numerical value is found to be

So,

If the function is squared and double integration is done on it. The square is values as .

The expression for area of a d-dimensional hypersphere is given as,

The volume of segment that is unit thick is given by,

Integrating this expression,

This is the formula for the volume of a d-dimensional hypersphere.