Sometimes, we need a precise value for following
As such, I am going to calculate it and place the result here for future works.
In order to evaluate the above integral, we need to use the so-called co-area formula. We first write
where we have used
In particular, by the duplication formula, we get that
where is the volume of -sphere of radius 1 in .
Another particular formula occurs when . We then have
where is the volume of -ball of radius 1 in .
Keep in mind that the -sphere of radius 1 is nothing but the boundary the -ball of radius 1. Therefore, is nothing but the surface area of the -ball of radius 1. Interestingly, we have the following well-known formula
As such, the previous formula can be expressed as