S is the upper cap cut from the sphere by the cylinder

Ok, so

on the xy-plane[/tex]

P is a vector normal to R, so p=k

So, now I have

After factoring out constants and rewriting z as a function of x and y I have

Now, converting to polar coordinates, I get

Ok, I let . Then,

Which then gives me

Then I get

Now, I have

So

The answer in the book is . can someone help me find my mistake(s)? Thanks.