What you are doing, apparently, is projecting the surface of the sphere onto the xy-plane but once you have projected to the xy-plane, there is no longer a variable. You are using polar coordinates with r from 0 to R and from 0 to . Be careful to note that the both the "top" and "bottom" hemispheres project to the same disk in the xy-plane.

Personally, I would use the fact that the "vector differential of surface area" for a sphere of radius R is and integrate with from 0 to , from 0 to .