Asymptotic volume of n-sphere intersection with n-cube

I am looking for an lower bound on the following quantity in terms of $d, r_1, r_2$:$$\text{Vol}(C^d(r_1) \cap B^d(r_2))$$where $B^d(r) = \{(x_1, \dots, x_d) \mid x_1^2 + \dots + x_d^2 = r^2\}$ and $C^d(r) = [-r, r]^d$. Specifically, I am interested in asymptotic bounds as $d \to \infty$. The key here is that $r_1 \neq r_2$, so the corners of the cube will "stick out" from the sphere. If $r_1 = r_2$ this is just the volume of the sphere However, I am not sure how to bound the quantity for other values of $r_1, r_2$.