I always have to draw a graph for these - or at least imagine one. You have a parabola and a horizontal line; the intersections are (-1,1) and (1,1). You get to choose whether to integrate y first or x first - I would have gone with y first, but it looks like you're doing x first. The right and left boundaries are the parabola, so solving for x gives and . Then y goes from 0 to 1, as you said.
If you integrate with respect to y first, then the top and bottom boundaries are the line and the parabola, so y goes from to 1. And then x goes from -1 to 1.