I suggest you switch to spherical coordinates as soon as possible. Review the section in your book on spherical coordinates. In particular, you should know how to describe half a sphere and that the Jacobian in spherical coordinates is $\displaystyle J(r, \theta, \varphi)=r^2\sin\theta$. In this case you'll have $\displaystyle 0\leq \theta\leq \pi/2$ (assuming the ball is as you've graphed it).
Spherical coordinate system - Wikipedia, the free encyclopedia
TheEmptySet told you where the error lies. I used the symbols from the Wikipedia article where a unit sphere is $\displaystyle 0 < r < 1, 0 < \theta < \pi, 0 < \varphi < 2\pi$.
You have switched the limits for $\displaystyle \theta$ and $\displaystyle \varphi$.
I was referring to the unit sphere.
What you posted is only correct if the Jacobian for the spherical coordinates in this problem is $\displaystyle r^2\sin\varphi$. Again, read the Wikipedia article I linked to if you're confused about the symbols used.