There are a number of ways you can go about this: you can use the stuff in vector calculus or you can set up an area integral where you map the surface to what is called a chart and all this means is that you take the surface and stick it on a flat piece of paper.
Mathematically this is just a transformation or a change of variables where your two variables are the north/south line and the east-west line where you are trying to find the area as if you took your surface and put it on a flat piece of paper like the following:
So you can either transform your surface in its local system (r,theta,phi) to something like the above and use regular integration (finding the area of the region) or you can use vector calculus and use what is called a surface integral. Both do the same thing and the surface integral actually does this projection for you by using the dot or inner products (which are just ways of finding specific kinds of projection components).
Are you currently doing vector calculus?