$(50$ points$)$ The yellow area shown below is defined at the bottom by the function $y=\frac{1}{64}{x}^3,$ at the left by the $y-$axis and on the top by a horizontal line at $y=8mm.$ Find the moments of inertia and product of inertia about the x $-$ and y $-$axes and the polar moment of inertia. Also find the radii of gyration with regards to the x $-$ and y $-$axes and the origin.

$($Result check : $I_x=\frac{6,144}{5}m{m}^4,I_y=\frac{2,048}{3}m{m}^4,I_{xy}=768m{m}^4,J_O=\frac{28,672}{15}m{m}^4,(k_x=\frac{\sqrt[\phantom{2}]{10,190}}{20}(mm),k_y=\frac{8}{3}\sqrt[\phantom{2}]{2}(mm),k_O=\frac{16}{15}\sqrt[\phantom{2}]{35}(mm).\}$