So suppose we have to calculate $$\iiint V\, \mathrm dx\,\mathrm dy\,\mathrm dz,$$ where $z=6-x^2-y^2$, $z^2=x^2+y^2$, $z=0$.
I plugged the equations in GeoGebra so I have a fairly good concept of what the required area is, but the solution of the integral states that we have to divide the area into two sub-areas $V_1$ and $V_2$, so that:
$V_1 : 0 \le z \le \sqrt{x^2+y^2}$
$V_2 : 0 \le z \le 6-x^2-y^2$
However, I fail to see why we need to break up the integral in the first place. Why do we do that? Couldn't we just have introduced cylindrical coordinates for the whole integral?