Let's consider a two-sided an unit ice cream cone defined by $E = \left \{ (x,y,z): x^2 + y^2 \leq z^2 \leq 1 - x^2 - y^2 \right \}$
What is volume of this icecream cone (I only want one side because the other side is going to fall to the ground)?
Under cylindrical coordinates, the ice cream cone's volume is
$\int_{0}^{2\pi} \int_{0}^{1} \int_{r}^{\sqrt{1-r^2}}rdz dr d\theta $
Now for some reason this integral gives me $0$
Under spherical coordinates, the ice cream cone's volume
$\int_{0}^{2\pi}\int_{0}^{\frac{\pi}{4}}\int_{0}^{1}\rho^2 \sin\phi d\rho d\phi d\theta$
And there is no way this is $0$.
Could someone tell me what's wrong with the first integral?
Thank you