One could integrate over y by using
; however, when you put in the limits, the resulting integral over x becomes very difficult.
Note that the region of integration, the region bounded by the parabolas and , has twofold rotational symmetry about the origin. Thus, we consider the change in variables , which represents axes rotated by 180°. We then have
and if we rename u and v as x and y, respectively, we see
For our integrand , we have , which combined with the rotational symmetry of the region of integration, means that the integral is zero.
Monday Math 148