CALC $3$ EXPERT USING INTEGRALS

$3.$ A spherical shell of radius $80$ cm is constructed with the top half made of gold with density $1\mathrm{.}86\backslash (\backslash$mathrm$\{$mg$\}/\backslash$mathrm$\{$cm$\}^\{2\}\backslash )$ and bottom half made of copper with density $\backslash (\mathrm{.}86\backslash$mathrm$\{$mg$\}/\backslash$mathrm$\{$cm$\}^\{2\}\backslash ).$

a$)$ Find the center of mass of the shell.

b$)$ Find the moment of inertia about a diameter of the shell, where the top half of the diameter coincides with the gold half of the shell and the bottom half coincides with the copper half of the shell.

c$)$ Find the moment of inertia about a diameter of the shell, where the gold half of the shell is above the diameter and the copper half of the shell is below the diameter.

Warning: this is a shell not a solid sphere, you need to use surface integrals. I would suggest picking your coordinate system with origin at the center of the shell, and the gold half corresponding to a positive $\backslash ($ z $\backslash )$ and the copper half corresponding to a negative $\backslash ($ z $\backslash ).$