Answer
The description in cylindrical coordinates is
$\left\{ {\left( {r,\theta ,z} \right)|r \le 2,0 \le \theta \le 2\pi } \right\}$
The description in spherical coordinates is
$\left\{ {\left( {\rho ,\theta ,\phi } \right)|\rho \sin \phi \le 2,0 \le \theta \le 2\pi ,0 \le \phi \le \pi } \right\}$
Work Step by Step
In cylindrical coordinates, we have ${r^2} = {x^2} + {y^2}$. Since ${x^2} + {y^2} \le 4$, so ${r^2} \le 4$. Thus, the description in cylindrical coordinates is
$\left\{ {\left( {r,\theta ,z} \right)|r \le 2,0 \le \theta \le 2\pi } \right\}$
In spherical coordinates, we have
$x = \rho \sin \phi \cos \theta $
$y = \rho \sin \phi \sin \theta $
So,
${x^2} + {y^2} = {\rho ^2}{\sin ^2}\phi {\cos ^2}\theta + {\rho ^2}{\sin ^2}\phi {\sin ^2}\theta $
${x^2} + {y^2} = {\rho ^2}{\sin ^2}\phi \left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)$
Since ${\cos ^2}\theta + {\sin ^2}\theta = 1$, so ${x^2} + {y^2} = {\rho ^2}{\sin ^2}\phi$.
Since ${x^2} + {y^2} \le 4$, so ${\rho ^2}{\sin ^2}\phi \le 4$. Thus, the description in spherical coordinates is
$\left\{ {\left( {\rho ,\theta ,\phi } \right)|\rho \sin \phi \le 2,0 \le \theta \le 2\pi ,0 \le \phi \le \pi } \right\}$
