Surface area of intersecting cylinder and sphere

[andrewjb]

I have a problem that I've been stuck on for a while as follows,

Find the surface area of the part of the cylinder x^{2}+y^{2}=2ay in the first octant that lies inside the sphere x^{2}+y^{2}+z^{2}=4a^{2}. Express your answer in terms of a single integral in \phi, you do not need to evaluate this integral.


I've started by parametrization the cylinder as S(\theta,z)=(\sqrt{2ay}cos(\theta),(\sqrt{2ay}sin(\theta),z). I then went on take the derivative of S in terms of \theta and z and took the cross product of the terms. I know the bounds of integration for \theta should be 0 to Pi/2, but from there I'm unsure of what to do in terms of setting up the bounds for z.

Any help would be appreciated, thanks!

 

[Defennder]

That looks more like an ellipse than a circle in the x-y plane. There is a special parametrisation for ellipses, but you have to express the equation in the form of an ellipse first. Secondly, your parametrisation for S(\theta,z) is incorrect. It is supposed to consist only of \theta,z and other constants. It shouldn't have y as a variable.