Answer
The description of the given set in spherical coordinates:
$0 \le \rho \le 1$, ${\ \ }$ $\theta = \frac{\pi }{4}$.
Work Step by Step
In spherical coordinates:
1. the radial coordinate $\rho $
Since ${\rho ^2} = {x^2} + {y^2} + {z^2}$, so we have ${\rho ^2} \le 1$. Since $\rho $ is always positive, we have $0 \le \rho \le 1$.
2. the angular coordinate $\theta$ satisfies
$\tan \theta = \frac{y}{x} = \frac{y}{y} = 1$, ${\ \ }$ $\theta = \frac{\pi }{4},\frac{{5\pi }}{4}$
Since $x \ge 0$, $y \ge 0$, it is constrained in the first quadrant. So,the right choice is $\theta = \frac{\pi }{4}$.
3. the angular coordinate $\phi$
There is no constraint on $\phi$.
Therefore, the description of the given set in spherical coordinates is
$0 \le \rho \le 1$, ${\ \ }$ $\theta = \frac{\pi }{4}$.
