find the volume bounded above by the sphere

[spdrmncoo]

I would like some help with this problem, please.

Find the volume bounded above by the sphere p = 6^(1/2) and below by the paraboloid z = x^2 + y^2. Locate the centroid of this region (x, y, z).

 

[galactus]

Using cylindrical coordinates:

The equation of the sphere \(\displaystyle \L\\{\rho}=\sqrt{6}\) would be \(\displaystyle \L\\x^{2}+y^{2}+z^{2}=6\).

Since \(\displaystyle \L\\x^{2}+y^{2}=r^{2}\)

We have(I hope):

\(\displaystyle \L\\\int_{0}^{2{\pi}}\int_{0}^{1}\int_{r^{2}}^{\sqrt{6-r^{2}}}rdzdrd{\theta}\)

 

[spdrmncoo]

thank you