I'm trying to integrate this using spherical coordinates (this is the only information given by the way). My issue is understanding how to find the range of $φ$ and $θ$. I know that $0≤ρ≤3$. But for $φ$ and $θ$, I've watched numerous videos but I still fail to understand how to find the coordinates on a general level. Any help please? Thanks in advance
Note that if we intersect the volume of integration with the $xy$-plane, then we obtain a quarter-circle of radius $3$ located in Quadrant IV (that is, $x \geq 0$ and $y \leq 0$): \begin{align*} S &= \{(x, y) \mid x \in [0, 3] \text{ and } y \in [-\sqrt{9 - x^2}, 0]\} \\ &= \{(r, \theta) \mid r \in [0, 3] \text{ and } \theta \in [\tfrac{3\pi}{2}, 2\pi]\} \end{align*}
As for $\varphi$, imagine intersecting the volume of integration with an arbitrary half plane $\theta$ to obtain an "$rz$-half-plane" (since $r \geq 0$). Then since: $$ z \in [0, \sqrt{9 - x^2 - y^2}] = [0, \sqrt{9 - r^2}] $$ it follows that we have a quarter-circle of radius $3$ located in Quadrant I (that is, $r, z \geq 0$). Hence, since $\cos\varphi = \tfrac{z}{\rho} \in [0, 1]$, it follows that $\varphi \in [0, \tfrac{\pi}{2}]$.