1. **State the problem:** We are given the function $y = \tan(x)$ and a region $R$ bounded by the curve $y = \tan(x)$, the x-axis, and the vertical line $x = \frac{\pi}{6}$. We want to approximate the area of $R$ using the trapezium rule with the given table values, then find the exact area by integration, and finally calculate the percentage error. 2. **Given data:** \begin{align*} x &: 0, \frac{\pi}{24}, \frac{\pi}{12}, \frac{3\pi}{24}, \frac{\pi}{6} \\ y &: 0, \text{Box 1}, \text{Box 2}, \text{Box 3}, 0.577 \end{align*} We know $y = \tan(x)$, so calculate the missing values: \begin{align*} \tan\left(\frac{\pi}{24}\right) &= \tan(7.5^\circ) \approx 0.132 \\ \tan\left(\frac{\pi}{12}\right) &= \tan(15^\circ) \approx 0.268 \\ \tan\left(\frac{3\pi}{24}\right) &= \tan(22.5^\circ) \approx 0.414 \end{align*} 3. **Apply the trapezium rule:** The trapezium rule for $n$ intervals with points $x_0, x_1, ..., x_n$ and function values $y_0, y_1, ..., y_n$ is: $$\text{Area} \approx \frac{h}{2} \left(y_0 + 2\sum_{i=1}^{n-1} y_i + y_n\right)$$ where $h$ is the width of each subinterval. Here, $h = \frac{\pi}{24}$ and $n=4$ intervals. Calculate: $$\text{Area} \approx \frac{\pi/24}{2} \left(0 + 2(0.132 + 0.268 + 0.414) + 0.577\right)$$ Sum inside parentheses: $$2(0.132 + 0.268 + 0.414) = 2(0.814) = 1.628$$ Total: $$0 + 1.628 + 0.577 = 2.205$$ So area approximation: $$\text{Area} \approx \frac{\pi}{48} \times 2.205 \approx 0.1445 \times 2.205 = 0.318$$ 4. **Exact area by integration:** The exact area under $y = \tan(x)$ from $0$ to $\frac{\pi}{6}$ is: $$\int_0^{\pi/6} \tan(x) \, dx$$ Recall: $$\int \tan(x) \, dx = -\ln|\cos(x)| + C$$ Evaluate: $$\int_0^{\pi/6} \tan(x) \, dx = [-\ln|\cos(x)|]_0^{\pi/6} = -\ln\left(\cos\frac{\pi}{6}\right) + \ln(\cos 0)$$ Calculate values: $$\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2} \approx 0.866, \quad \cos 0 = 1$$ So: $$= -\ln(0.866) + \ln(1) = -(-0.1447) + 0 = 0.1447$$ Rounded to 3 decimal places: $$0.145$$ 5. **Percentage error:** $$\text{Percentage error} = \left|\frac{\text{Approximate} - \text{Exact}}{\text{Exact}}\right| \times 100 = \left|\frac{0.318 - 0.145}{0.145}\right| \times 100 = 119\%$$ Rounded to nearest integer: 119% **Final answers:** - Approximate area (trapezium rule): $0.318$ - Exact area (integration): $0.145$ - Percentage error: $119\%$