Chapter 15: Multiple Integrals - Section 15.6 - Moments and Centers of Mass - Exercises 15.6 - Page 908: 8

$$\frac{\pi}{2}$$

\begin{align*} I_{y}&=\int_{\pi}^{2 \pi} \int_{0}^{\left(\sin ^{2} x\right) / x^{2}} x^{2} d y d x\\ &=\int_{\pi}^{2 \pi}\left(\sin ^{2} x-0\right) d x\\ &=\frac{1}{2} \int_{\pi}^{2 \pi}(1-\cos 2 x) d x\\ &=\frac{1}{2} \left(x- \frac{1}{2}\sin 2x\right)\bigg|_{\pi}^{2 \pi} \\ &=\frac{\pi}{2} \end{align*}