Octant issue with Sphere, and Elliptic Paraboloid

Question

This is the graphical setup for my problem Math3d. I given a sphere in cylindrical coordinates as the following \begin{equation}r^2+z^2=20 \end{equation} Which is the equation of a sphere of radius $\rho=2\sqrt{5}$. The problem has to be solved in the following manner (which is cylindrical systems): \begin{equation}\iiint_G f(r,\theta,z)dzdrd\theta \end{equation} The equation for the elliptical paraboloid is the following in cylindrical coordinates system. \begin{equation}z=r^2\end{equation} The problem that needs to be solved at hand is to find the volume inside the surface: \begin{equation} r^2+z^2=20 \end{equation} The volume is constrained to no being above the following surface as well: \begin{equation}z=r^2\end{equation}The question I have involves in how I set up my integral which was the following: \begin{equation} \int_0^{2\pi} \int_0^2 \int_0^{r^2} r^3dz dr d\theta \end{equation} I had hoped whether it was correct, however, there were doubts, but alas I shall post my own answer.

Answer 1

The following integral setup solves for that particular problem is the following:\begin{equation} \int_0^{2\pi}\int_2^{2\sqrt{5}}\int_{-\sqrt{20-r^2}}^{\sqrt{20-r^2}}rdzdrd\theta+\int_0^{2\pi}\int_0^2\int_{-\sqrt{20-r^2}}^{r^2}rdzdrd\theta\end{equation} The result is the following which is \begin{equation}\iiint_Gf(r,\theta,z)dzdrd\theta=\frac{80\sqrt5+152}{3}\pi\end{equation}