A) Find the volume bounded by sphere rho=rt(6) and the paraboloid
z=x^2 + y^2
B) Locate the centroid of this region
Any help, tips, work, or similar problems would be greatly appreciated. =)
The upper surface (sphere) is given by $\displaystyle z=\sqrt{36 - x^2-y^2}$.
Thus, the volume is,
$\displaystyle \int_A \sqrt{36-x^2-y^2} - (x^2+y^2) \ dy~dx $ where $\displaystyle A$ is the disk of radius $\displaystyle 6$.
Using polar change of variable,
$\displaystyle \int_0^{2\pi} \int_0^6 (\sqrt{36-r^2} - r^2)rdr~d\theta$