So I'm asked to evaluate the integral by changing to spherical coordinates, $\displaystyle \displaystyle \int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{\sqrt{x^{2} + y^{2}}}^{\sqrt{2 - x^{2} - y^{2}}}xy \, dz \, dy \, dx$.

I see how do to do all the plugging in, but I'm not sure how to determine the limits of integration. I know my radius is going to start at 0, but I want it to be limited by some function of $\displaystyle \phi$. I can see that I want to be looking only at what's above the xy-plane and toward the positive side of x, so I want $\displaystyle -\pi/2 \leq \theta \leq \pi/2$ and $\displaystyle \pi/2 \leq \phi \leq \pi$. So I guess all that I don't see is what function bounds the radius on top.