Answer
$\mathop \smallint \limits_{\theta = 0}^\pi \mathop \smallint \limits_{r = 0}^1 \mathop \smallint \limits_{z = 0}^{{r^2}} f\left( {r\cos \theta ,r\sin \theta ,z} \right)r{\rm{d}}z{\rm{d}}r{\rm{d}}\theta $
Work Step by Step
We have $\mathop \smallint \limits_{x = - 1}^1 \mathop \smallint \limits_{y = 0}^{\sqrt {1 - {x^2}} } \mathop \smallint \limits_{z = 0}^{{x^2} + {y^2}} f\left( {x,y,z} \right){\rm{d}}z{\rm{d}}y{\rm{d}}x$.
From the order of the integration, we obtain the region description:
${\cal W} = \left\{ {\left( {x,y,z} \right)| - 1 \le x \le 1,0 \le y \le \sqrt {1 - {x^2}} ,0 \le z \le {x^2} + {y^2}} \right\}$
Notice that this is a $z$-simple region such that the projection of ${\cal W}$ onto the $xy$-plane is the upper half of the disk ${x^2} + {y^2} \le 1$, located in the first and the second quadrant. Since in cylindrical coordinates, we have ${r^2} = {x^2} + {y^2}$, so the description of ${\cal W}$:
${\cal W} = \left\{ {\left( {r,\theta ,z} \right)|0 \le r \le 1,0 \le \theta \le \pi ,0 \le z \le {r^2}} \right\}$
So, the triple integral in cylindrical coordinates:
$\mathop \smallint \limits_{\theta = 0}^\pi \mathop \smallint \limits_{r = 0}^1 \mathop \smallint \limits_{z = 0}^{{r^2}} f\left( {r\cos \theta ,r\sin \theta ,z} \right)r{\rm{d}}z{\rm{d}}r{\rm{d}}\theta $
