A farmer constructs a trough by welding a rectangular piece of $2-$mm$-$thick sheet steel to half of a steel drum. Knowing that the density of steel is $\backslash (7850\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash )$ and the thickness of the walls of the drum is $1\mathrm{.}8$ mm $,$ determine the mass moment of inertia of the trough with respect to each of the coordinate axes. Neglect the mass of the welds.

The mass moment of inertia of the component with respect to $\backslash ($ x $\backslash )$ axis is $\backslash (\backslash$mathrm$\{$kg$\}\backslash$cdot $\backslash$mathrm$\{$m$\}^\{2\}\backslash ).$

The mass moment of inertia of the component with respect to $\backslash ($ y $\backslash )$ axis is $\backslash (\backslash$mathrm$\{$kg$\}\backslash$cdot $\backslash$mathrm$\{$m$\}^\{2\}\backslash ).$

The mass moment of inertia of the component with respect to $\backslash ($ z $\backslash )$ axis is $\backslash (\backslash$mid $\backslash$mathrm$\{$kg$\}\backslash$cdot $\backslash$mathrm$\{$m$\}^\{2\}\backslash ).$