Answer
$\rho=1$ , $0\leq \phi\leq \pi/2$, $0\le \theta \le 2\pi$.
Work Step by Step
The set $x^2+y^2+z^2=1$, $z\geq 0$ is the upper hemisphere in $R^3$, centered at the origin. Since $$
\begin{aligned}
&x=\rho \sin \phi \cos \theta\\
&y=\rho \sin \phi \sin \theta\\
&z=\rho \cos \phi,
\end{aligned}
$$we have $$x^2+y^2+z^2=\rho^2(\sin^2\phi\cos^2\theta+\sin^2\phi\sin^2\theta+\cos^2\phi)\\ =\rho^2$$ That is, $\rho^2= 1$ and since $\rho$ is non negative then $\rho=1$ and since $z\geq 0$, then $\rho\cos \phi \geq0$ i.e, $0\leq \phi\leq \pi/2$.
There are no restrictions on $\theta$, so: $0\le \theta \le 2\pi$
