$4\mathrm{.}4($a$)$ Using integration, find the moment of inertia about the $x$ axis for a thin semi$-$circular arc of radius $R$ where the thickness $t(R).$

$($b$)$ Dividing a semi$-$circular area of radius $R$ into semi$-$circular arcs, and using the result of $(a),$ find the moment of inertia of a semi$-$circular area about its diameter.