We can also write d(x,y) = 12x. Does that make it look more doable?

I think you'll want to integrate with respect to y first, because then the innermost integrand will just be a constant (with respect to y). The limits on your integral should be

outer: 0 to 1

inner: x^2 to x

EDIT: I was thinking in a probability distribution mode. I'll have to see how well it applies to center of mass. Anyway the integral limits should be fine.