How can I find the ratio of the volume of a region in the unit n-dimensional hypersphere where for a radius $0<r<1$, the region is bounded such that
$r^2<\sum_{i=\{1...n\}}x_i^2<1$ ; $\sum_{j=\{1...m\}}x_j^2<r^2$ for a given $m\in\{1...n-1\}$.
I've been trying to think about this problem but I am finding it hard relating the volume of the condition $\sum_{j=\{1...m\}}x_j^2<r^2$ to the volume of the area $r^2<\sum_{i=\{1...n\}}x_i^2<1$.