1. **State the problem:** We have the function $y=\sin(x^2)$ and need to fill in missing $y$-values at given $x$ points, then approximate the area of region $R$ bounded by the curve, $x$-axis, and vertical lines $x=\frac{1}{2}$ and $x=\frac{3}{2}$ using the trapezium rule. 2. **Calculate missing $y$-values:** - Box 1: $y=\sin\left(\left(\frac{1}{2}\right)^2\right)=\sin\left(\frac{1}{4}\right)$ - Box 2: $y=\sin\left(\left(\frac{3}{4}\right)^2\right)=\sin\left(\frac{9}{16}\right)$ - Box 3: $y=\sin\left(\left(\frac{5}{4}\right)^2\right)=\sin\left(\frac{25}{16}\right)$ - Box 4: $y=\sin\left(\left(\frac{3}{2}\right)^2\right)=\sin\left(\frac{9}{4}\right)$ 3. **Evaluate these values to 3 decimal places:** - Box 1: $\sin(0.25) \approx 0.247$ - Box 2: $\sin(0.5625) \approx 0.534$ - Box 3: $\sin(1.5625) \approx 0.999$ - Box 4: $\sin(2.25) \approx 0.778$ 4. **Apply trapezium rule:** The trapezium rule for $n=4$ intervals with points $x_0=\frac{1}{2}, x_1=\frac{3}{4}, x_2=1, x_3=\frac{5}{4}, x_4=\frac{3}{2}$ and corresponding $y$ values $y_0, y_1, y_2, y_3, y_4$ is: $$\text{Area} \approx \frac{h}{2} \left(y_0 + 2y_1 + 2y_2 + 2y_3 + y_4\right)$$ where $h = x_{i+1} - x_i = \frac{1}{4} = 0.25$. 5. **Substitute values:** $$\text{Area} \approx \frac{0.25}{2} \left(0.247 + 2(0.534) + 2(0.841) + 2(0.999) + 0.778\right)$$ Calculate inside the parentheses: $$0.247 + 1.068 + 1.682 + 1.998 + 0.778 = 5.773$$ 6. **Calculate area:** $$\text{Area} \approx 0.125 \times 5.773 = 0.7216$$ Rounded to 3 decimal places: $$\boxed{0.722}$$ **Final answers:** - Box 1 = 0.247 - Box 2 = 0.534 - Box 3 = 0.999 - Box 4 = 0.778 - Area of $R$ using trapezium rule $\approx 0.722$