Hi, just need a quick hand on an assignment question:

What I've done so far is (convert to spherical) parameterise

by

And taken the vectors

and

, found their cross product as a normal vector, then taken the dot product with

.

What results is something horrible (seeing as the integral must be taken), and it makes me think that something has gone very wrong:

Obviously I'm not looking for an answer to the problem, but I'm pretty sure that I've made an error somewhere, and I would appreciate it if someone could point it out. More details of the in-between steps can be supplied, if needed.

Thanks in advance.