In a region bounded below by plane $z=0$, above by the sphere $x^2+y^2+z^2=4$ and on the sides by the cylinder $x^2+y^2=1$, set up the triple integrals in spherical coordinates that give the volume of the region using following order of integrations.
$d\rho\ d\phi\ d\theta$
$d\phi\ d\rho\ d\theta$
If $\rho$ is written first then I interpret it as follows:
For first its ok to see that $\rho$ starts from $0$ but it will end on two distinct surfaces namely cylinder and sphere corresponding values of $\phi$ are then shown in the integrals$$\underbrace{\int_{0}^{2\pi}\int_0^{\frac{\pi}{6}}\int_0^2\rho^2\sin\phi\cdot d\rho\ d\phi\ d\theta}_\color{blue}{\text{sphere}}+\underbrace{\int_{0}^{2\pi}\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\int_0^{\csc\phi}\rho^2\sin\phi\cdot d\rho\ d\phi\ d\theta}_\color{purple}{\text{cylinder}}$$
For the second one answer provided is $$\bigg(\int_{0}^{2\pi}\int_1^{2}\int_{\frac{\pi}{6}}^{\sin^{-1}(1/\rho)}+\int_{0}^{2\pi}\int_{0}^{2}\int_0^{\frac{\pi}{6}}+\int_{0}^{2\pi}\int_{0}^{1}\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\bigg)\rho^2\sin\phi\cdot d\rho\ d\phi\ d\theta$$
How do I interpret this, please don't say try to write $\phi$ in therms of $\rho$ and $\theta$, I'm searching for more intuitive answers of writing limits.