Region of integration of intersection between cone and sphere 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:

 A: If I understand the geometric description, the center of the sphere can be chosen to lie at $(R\sin\alpha, 0, R\cos\alpha)$ (i.e., in the positive $x$-direction in the $(x, z)$-plane). The ball bounded by the sphere is the solution set of
$$
(x - R\sin\alpha)^{2} + y^{2} + (z - R\cos\alpha)^{2} \leq R^{2},
$$
or
$$
x^{2} + y^{2} + z^{2} \leq 2xR\sin\alpha + 2zR\cos\alpha.
$$
Using spherical coordinates
$$
x = \rho \cos\theta \sin\phi,\qquad
y = \rho \sin\theta \sin\phi,\qquad
z = \rho \cos\phi,
$$
in which $\theta$ denotes longitude and $\phi$ denotes colatitude, and canceling a common factor of $\rho$, the ball is described by
$$
\rho \leq 2R(\cos\theta \sin\phi \sin\alpha + \cos\phi \cos\alpha).
$$
The bounds on the solid region of intersection are therefore
$$
0 \leq \theta \leq 2\pi,\qquad
0 \leq \phi \leq \alpha,\qquad
0 \leq \rho \leq 2R(\cos\theta \sin\phi \sin\alpha + \cos\phi \cos\alpha).
$$
If you really did mean "half a cone with vertex angle $\alpha$" (rather than a "cone with half-vertex angle $\alpha$", as I've assumed), this description will require modification (and more information: where is the center of the sphere on the half-cone?).
