Numerical integration over a hemisphere

Dear all,

I am trying to solve the following problem:

A function is given on the surface of a unity hemisphere, and I need to find an efficient way to numerically integrate it over the surface.

I started with an equidistant approach (both angles of the spherical coordinates subdivided into constant intervals). This gives good results but needs a large amount of computations.

In the internet, I found an alternative (Gaussian) approach with equidistant points, but only for integration over spheres, not over hemispheres. This approach can be found here: Integration nodes for the sphere

Can somebody help me on doing the numerical integration over the hemisphere? Any help is appreciated!

Thank you,

Jens