Answer
$$
I_y=2614 \text { slug } \cdot \mathrm{ft}^2
$$
Work Step by Step
The mass of the differential element is $d m=\rho d V=\rho\left(\pi y^2\right) d x=\rho \pi x d x$.
$$
\begin{aligned}
d I_y & =\frac{1}{2} d m y^2+d m x^2 \\
& =\frac{1}{4}[\rho \pi x d x](x)+(\rho \pi x d x) x^2 \\
& =\rho \pi\left(\frac{1}{4} x^2+x^3\right) d x \\
I_y & =\int d I_y=\rho \pi \int_0^4\left(\frac{1}{4} x^2+x^3\right) d x=69.33 \pi \rho \\
& =69.33(\pi)(12)=2614 \text { slug } \cdot \mathrm{ft}^2
\end{aligned}
$$
