Evaluate using spherical coordinates:
I can represent this in cartesian coordinates like this:
and from here convert to spherical coordinates. I'm not sure how to completely convert this to spherical coordinates.
The only thing I could see was that on the xy-plane, part of the area is a quarter of a circle in the first quadrant with radius 1 (since y goes from 0 to and x goes from 0 to 1), so would go from 0 to .