If you look at your integral, the limits are actually lines on the plane, right? Try using those as bounds and compare to where the parabola comes across them.

So you would have your parabola on the uv-plane bounded by:

. Something to note: since you have your integral limit set up as a to , your integral should reflect this with: . Order is important. This intersected area should be the region you're finding the surface area for.

Hope this helps, and if anywhere I am mistaken, someone please feel free to correct me and get him pointed in the right direction! =)