I don't follow your limits for the polar set up:

z is bounded by 0 and y (rsinθ)

r is bounded by 0 and 2

... shouldnt θ be bounded by 0 and π? I thought it was the top-half of the circle (not the part in the 1st quadrant)

and then...

$\displaystyle yz = rsin{\theta}z r dz dr d{\theta}$

This?

$\displaystyle -\sqrt{4-x^{2}} <= y <= 0$