Answer
The description of the given set in spherical coordinates is
$0 \le \rho \le 2$, ${\ \ }$ $\theta = \frac{\pi }{2}$ or $\theta = \frac{{3\pi }}{2}$.
Work Step by Step
In spherical coordinates:
1. the radial coordinate $\rho$
Since $x=0$ and ${\rho ^2} = {x^2} + {y^2} + {z^2}$, so we have ${\rho ^2} \le 4$. Since $\rho$ is always positive, we have $0 \le \rho \le 2$.
2. the angular coordinate $\theta$ satisfies
$\cos \theta = \frac{x}{r} = \frac{x}{{\sqrt {{x^2} + {y^2}} }}$
Since $x=0$, we have $\cos \theta = 0$. So, $\theta = \frac{\pi }{2},\frac{{3\pi }}{2}$
3. the angular coordinate $\phi$
There is no constraint on $\phi$.
Therefore, the description of the given set in spherical coordinates is
$0 \le \rho \le 2$, ${\ \ }$ $\theta = \frac{\pi }{2}$ or $\theta = \frac{{3\pi }}{2}$.
