[Math] Spherical coordinates and unregular paraboloid

Question

If we have the following equation of a paraboloid:
$z=4-x^2-y^2$ and we have the region in space bounded by this paraboloid from above and by the $xy$-plane from below; we want to find the volume of this region using the spherical coordinates in the following orders; $d\rho\,d\phi\,d\theta$,
$d\phi\,d\rho\,d\theta$.

In the first order if we follow a ray from the origin, it will cut the surface which is the one of the paraboloid but we will end up with a second degree polynomial equation…

Answer 1

From:

$$x^2+y^2=4-z$$

we obtain:

$$\rho^2 \sin^2\phi=4-\rho \cos \phi\implies\rho^2 \sin^2\phi+\rho \cos \phi-4=0$$

and thus

$$\rho=\frac{-cos \phi\pm \sqrt{\cos^2 \phi+16\sin^2 \phi}}{2\sin^2 \phi}\implies \rho=\frac{-cos \phi+ \sqrt{\cos^2 \phi+16\sin^2 \phi}}{2\sin^2 \phi}>0$$

$$ \iiint \; dV = \int_0^{2\pi} \int_{0}^{\pi/2}\int_0^{\frac{-cos \phi+ \sqrt{\cos^2 \phi+16\sin^2 \phi}}{2\sin^2 \phi}}\rho^2 \sin \phi \; d\rho d \phi d \theta $$