If you want to integrate over the SURFACE of a spherical cap that is positioned in the way it is on wikipedia, this is rather simple. since it has azimuthal symmetry you get a factor $2\pi$ and for the altitude one just needs to integrate from $0$ to $\theta$ where $\theta$ is the angle that refers to the position. Spherical cap
Now I need to find out how to do this, if the cap is not symmetric around the z-axis but somewhere on the sphere. So I am looking for the most general way to find the integration interval for integration over the surface of a spherical cap. (Notice, I just need the interval). Since the integrand is highly ugly and unsymmetric I cannot reduce it to this general case with azimuthal symmetry.
Therefore, does anybody know how to integrate over the surface of an arbitrarily positioned spherical cap?