Answer
$$
\begin{aligned}
& I_x=53.3 \text { slug } \cdot \mathrm{ft}^2 \\
& I_y=26.7 \text { slug } \cdot \mathrm{ft}^2
\end{aligned}
$$
Work Step by Step
Mass of differential element is $d m=\rho d V=\rho\left(\pi z^2\right) d y=2 \rho \pi y d y$.
$$
\begin{gathered}
m=20=\int_m d m=\int_0^2 2 \rho \pi y d y \\
20=4 \rho \pi \quad \rho=\frac{5}{\pi} \text { slug } / \mathrm{ft}^3 \\
d I_x=\frac{1}{4} d m z^2+d m\left(y^2\right) \\
=\frac{1}{4}[2 \rho \pi y d y](2 y)+[2 \rho \pi y d y] y^2 \\
=\left(5 y^2+10 y^3\right) d y \\
I_x=\int d I_x=\int_0^2\left(5 y^2+10 y^3\right) d y=53.3 \mathrm{slug} \cdot \mathrm{ft}^2 \\
d I_y=\frac{1}{2} d m z^2=2 \rho \pi y^2 d y=10 y^2 d y \\
I_y=\int d I_y=\int_0^2 10 y^2 d y=26.7 \mathrm{slug} \cdot \mathrm{ft}^2
\end{gathered}
$$
