So, I have a solid bounded by $z=\sqrt{x^2+y^2}, z=\sqrt{1-x^2-y^2}, z=2$
I had to parametrize it using spherical coordinates so I used $$\begin{cases} x(\rho, \theta, \phi)=\rho\cos(\theta)\sin(\phi)\\ y(\rho, \theta, \phi)=\rho\sin(\theta)\sin(\phi)\\ z(\rho, \theta, \phi)=\rho\sin(\phi) \end{cases}$$ And $$\begin{cases} \rho\sin(\phi)=\sqrt{\rho^2\cos^2(\theta)\sin^2(\phi)+\rho^2\sin^2(\theta)\sin^2(\phi)} \\ \rho\sin(\phi) = \sqrt{1-\rho^2\cos^2(\theta)\sin^2(\phi)-\rho^2\sin^2(\theta)\sin^2(\phi)} \\ \rho\sin(\phi)=2 \end{cases} \Rightarrow \begin{cases} \phi=\pi/4 \\\rho=1\\ \rho=2/\cos(\phi) \end{cases}$$ So $\{(\rho, \theta, \phi)\in\Re^3 : 1\le\rho\le2/\cos(\phi)\land 0 \le \theta \le 2\pi \land 0 \le \phi \le \pi/4\}$
Maybe it can't be done with spherical coordinates but I wonder, could that solid be parametrized so that no variable is limited by a function as $\rho$ is?