I have a ball $B$ (let's assume it's centered at $0$) of radius $r_B$ in $\mathbb{R}^n$ and an ellipsoid $E$ centered in $c \neq 0$ defined by the equation $$\sum_{i=1}^n a_i (x_i - c_i)^2 \leq r_B$$
I want to compute the volume $V$ of $B \cap E$, but I don't really know where to start. For Ball/Ball intersection, it can be done using the Gauss-Bonnet theorem as shown here: https://hal.inria.fr/inria-00409374/document , but it doesn't seem to generalize to Ball/Ellipsoid intersection, so I don't really know where to start.