A rectangular slab of the same size and arranged in the same way as that in the question 4 has a surface density that varies with position according to the function \(\sigma(x, y)=a_{0}+a_{1} x y\) where \(a_{0}=0.1510^{3} \mathrm{~kg} \mathrm{~m}^{-2}\) and \(a_{1}=0.6010^{3} \mathrm{~kg} \mathrm{~m}^{-4}\). Determine the moment of inertia with respect to the \(y\)-axis.

Question 4

A rectangular plate of density \(\sigma=1.2510^{2} \mathrm{~kg} \mathrm{~m}^{-2}\) is arranged with the longer side \(\ell=75\) \(\mathrm{cm}\) along the \(x\) axis of a Cartesian reference frame, and with the shorter side \(h=30 \mathrm{~cm}\) along the \(y\) axis. Calculate the moment of inertia \(J_{x}\) with respect to rotations about the \(x\)-axis and the moment of inertia \(J_{y}\) with respect to rotations about the \(y\)-axis.