I need to find the volume of region bounded above by sphere $\displaystyle x^2 +y^2 +z^2=4 $ and below by the inverted cone $\displaystyle z=-\sqrt{x^2 +y^2} , -\sqrt{2}\leqslant z \leqslant 0$ and also find the centre of gravity with density equalling constant.

I set up my bound and triple integral as

$\displaystyle \int_{0}^{(3\pi /4)}\int_{0}^{(2\pi)}\int _{0}^{\2} r^2\sin \varphi drd\theta d\varphi$

and after solving i got approx 28.60

using symetry, the centre of gravity is on the z axis and using the formula for centre of mass:

$\displaystyle 1/m\iiint z dV$

and subsituting above result and

$\displaystyle z = r\cos\varphi$

i obtained

$\displaystyle \iiint r\cos \varphi r^2\sin \varphi drd\theta d\varphi$

solving with bounds stated in first part i got

$\displaystyle \bar{z} = 0.22$

does this look correct? thanks for any help.