The equivalent integral to $\displaystyle \int_{-1}^{1}\int_{0}^{\sqrt{1-y^2}}\int_{x^2+y^2}^{\sqrt{x^2+y^2}}xyz \ \mathrm{d}z\mathrm{d}x\mathrm{d}y$
The limits I got were $\displaystyle \int_{-\pi/2}^{\pi/2}\int_{0}^{1}\int_{r^2}^{r}zr^3\cos\theta \sin \theta \ \mathrm{d}z\mathrm{d}r\mathrm{d}\theta$
Is this right?
Yes you have it right .
For the volume of the regeion you have the correct integral $$ \displaystyle \int_{-1}^{1}\int_{0}^{\sqrt{1-y^2}}\int_{x^2+y^2}^{\sqrt{x^2+y^2}}xyz \ \mathrm{d}z\mathrm{d}x\mathrm{d}y$$
And changing to cylindrical coordinate you have the correct integral. $$\displaystyle \int_{-\pi/2}^{\pi/2}\int_{0}^{1}\int_{r^2}^{r}zr^3\cos\theta \sin \theta \ \mathrm{d}z\mathrm{d}r\mathrm{d}\theta$$