Need some help with a line integral problem...

Problem: Let C be the curve of intersection of the upper hemisphere and the cylinder , oriented counterclockwise as viewed from high above the xy-plane. Evaluate .

From what I understand, we need a parametrization in terms of one variable. I found an intersection curve parametrization as:

And now I'm not sure how to proceed. I suspect I will need to evaluate this integral:

Assuming all of that is correct, I suppose my primary question is how to find the limits of integration.