The paraoloid crosses the plane z= 2 where . That projects to the xy-plane as a circle with center (0, 0) and radius . Since we are only working in the first quadrant, it is a quarter circle and we have . That is, x goes from 0 to and, for each x, y goes from 0 to . Of course, for each x and y, z goes from to 2.
The integral is .