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.