$2.($Semicircular beam$).$ A beam has a semicircular cross section with radius $\backslash ($ a $\backslash ).$ Its center is aligned to the coordinate origin, as in the figure.

$($a$)$ Find the second moment of inertia of the cross section $\backslash ($ I$\_\{0\}\backslash )$ about neutral plane located at the bottom of the semicircle $\backslash (($y$=0)\backslash ).$

$($b$)$ Find the $\backslash ($ y $\backslash )-$coordinate of the centroid, $\backslash ($ y$\_\{$c$\}\backslash ).$

$($c$)$ Find the second moment of inertia of the cross section $\backslash ($ I$\_\{$c$\}\backslash )$ about the plane at $\backslash ($ y$\_\{0\}\backslash ),$ which is parallel to the $\backslash ($ x z $\backslash )-$plane.