Evaluate the following surface integral of the given function on the given surface.

g(x,y,z)=z on the first-octant part of the sphere

My work so far:

The domain:

Here is the point where I get stuck. In all my previous problems my g(x,y,z) only contained x's and y's. So, if that were the case here, I would just convert everything to polar coordinates, plug in the bounds and solve the integral. In this case though, there is a z in the integrand so I'm not sure how to continue. Could someone please help me out of a jam? Thanks