/sci/ - Science & Math

I'm trying to use Stoke's Theorem to evaluate the work done along the curve in pic related. I found the curl of my vector function and I solved for [math] z [/math] to see where the intersection is between the paraboloid and the cylinder, but I'm unsure where to go from here.
I know I'm looking for a double integral, but I can't remember how to find my boundary conditions. I would parameterize, but I'm looking to find this strictly in terms of [math] x [/math] and [math] y [/math].