Can one explain why the following formula holds for a n-dimensional ball volume?
I can't seem to understand why we are integrating over $r^{d-1}$, and not $r^d$ for example.
Intuitively, I understand that we are doing a summation of the surface of a sphere with radius ranging from 0 to 1, thus receiving a ball.
Very simple: The $(n-1)$-dimensional volume element of the sphere of radius $r$ is $r^{n-1}$ times the volume element of the unit sphere. (Think about the formulas for $dA$ in polar coordinates and $dV$ in $3$-dimensional spherical coordinates.)