The following is a simple and beautiful proof, shown to me to my great delight while I was in high school, that the -ball of radius has -volume
Although I expect most readers will know it, I believe that everybody should see it. I don’t know the history of it, and would be interested in learning.
Suppose that the -ball of radius has -volume . Then, by considering linear transformations, the -ball of radius has -volume . Moreover, differentiating this with respect to should produce the surface -volume of the -sphere : thus we expect .
Now consider the integral
We will compute in two different ways. On the one hand,
On the other hand, the form of suggests introducing a radial coordinate . Computing this way,