How to describe this region in spherical coordinates?

Question

How can you describe the region $B$ in spherical coordinates? $$B=\{(x,y,z)\in\Bbb R^3:x^2+y^2+z^2\leq 9,\ x^2+y^2\leq z^2\ z\geq 0\}$$ region equation

Answer 1

First, remember that $x^2+y^2+z^2 = r^2$, so the first equation becomes $r^2 \leq 9$, or $r < 3$. The second equation comes out of $x = r \sin \theta \cos \varphi$, $y = r \sin \theta \sin \varphi$ and $z = r \cos \theta$ (all important identities to memorize). Thus, we get $x^2 + y^2 = r^2 \sin^2 \theta$ and $z^2 = r^2 \cos^2 \theta$. Therefore, we have $r^2 \sin^2 \theta \leq r^2 \cos^2 \theta$, or $\tan^2 \theta \leq 1$. The last equation should be self explanatory from earlier statements

Answer 2

Reacalling that $$ \begin{align} x&=\rho\sin\phi\cos\theta\;,\\ y&=\rho\sin\phi\sin\theta\;,\\ z&=\rho\cos\phi\;, \end{align} $$ with $\rho\geq 0$, $\theta\in [0,2\pi)$ and $\phi\in [0,\pi]$, then \begin{align}&x^2+y^2+z^2\leq 9\Leftrightarrow \rho^2\leq 9\Leftrightarrow \rho\leq 3\\ &x^2+y^2\leq z^2\Leftrightarrow \rho^2\sin^2\phi\leq \rho^2\cos^2\phi\Leftrightarrow |\tan\phi|\leq 1\\ &z\geq 0\Leftrightarrow \rho\cos\phi\geq 0\Leftrightarrow \cos\phi\geq 0\end{align} Can you take it from here?