Let's suppose we have a sphere with radius $R>0$, and half a cone with an opening angle $\alpha \in (0, \pi/4)$. The vertex of the cone is in the surface of the sphere, and the center of the sphere is in the surface of the cone.
How can I find the region of integration of their intersection?
I have placed the cone so that its axis is in the direction of the Z axis, and its vertex is $(0, 0, 0)$. I think spherical coordinates will make things easier, but I don't know how to determine the region of integration.
Here's the region of integration, and how the cone and sphere are placed: