Parametrisation of a surface and a cylinder

Question

I have been asked to find C which is the curve I need to integrate over and C is the intersection of the cylinder $x^2+y^2=2y$ and the plane $y=z$. I assume you have to find a parametrization that will work for both of these surfaces but I don't know what would be a good parametrization to use. Help would be appreciated. Thanks

Answer 1

First of all,

$x^{2}+y^{2}=2y \implies x^{2}+(y-1)^{2}=1$

Then $(x,y)=(\cos t,1+\sin t)$.

Together with $y=z$, we have $$(x,y,z)=(\cos t,1+\sin t,1+\sin t)$$

which is an ellipse.