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?
You can rotate the vector field to match the re-positioning of the cap. Just multiply the field by a rotation matrix:
$$\vec{f}'(x,y,z)=R\vec{f}(x,y,z)$$
Then integrate $\vec{f}'(x,y,z)$ over the cap symmetric about the z-axis.