Let's solve the step by step both the **apezoidal Rule** and **Simpson's Rule**, using the graph and \( n = 6 \) (which means 7 points). ### **Step 1: Extract the y-values from the graph** have \( x \) values at \( , 1, 2, 3, 4, 5, 6 \). Approximate \( y \) values from the graph at each \( x \): - \( f() = 1 \) - \( f(1) = 3 \) - \( f(2) = 5 \) - \( f(3) = 3 \) - \( f(4) = 1 \) - \( f(5) = \) - \( f(6) = 2 \) **Explanation:** These \( y \)-values are visually estimated based on the graph's intersection with each vertical grid line. --- ### **Step 2a: Trapezoidal Rule Calculation** The Trapezoidal Rule formula: \[ T_n = \frac{h}{2} [f(x_) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + 2f(x_5) + f(x_6)] \] Where \( h = \frac{b - a}{n} = \frac{6 - }{6} = 1 \). Plug in values: \[ T_6 = \frac{1}{2} [1 + 2(3) + 2(5) + 2(3) + 2(1) + 2() + 2] \] \[ = \frac{1}{2} [1 + 6 + 10 + 6 + 2 + + 2] \] \[ = \frac{1}{2} [27] \] \[ = 13.5 \] **Explanation**: Each interior point is multiplied by 2, endpoints by 1, sum, then multiply by \( \frac{h}{2} \). --- ### **Step 2b: Simpson's Rule Calculation** Simpson's Rule formula (for even \( n \)): \[ S_n = \frac{h}{3} [f(x_) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + f(x_6)] \] With \( h = 1 \): Plug in values: \[ S_6 = \frac{1}{3} [1 + 4(3) + 2(5) + 4(3) + 2(1) + 4() + 2] \] \[ = \frac{1}{3} [1 + 12 + 10 + 12 + 2 + + 2] \] \[ = \frac{1}{3} [39] \] \[ = 13 \] **Explanation:** Alternate coefficients 4 and 2 for interior points, endpoints by 1, sum, then multiply by \( \frac{h}{3} \). --- ## **Final Answers** - **Trapezoidal Rule estimate (\(T_6\))**: **13.5** - **Simpson's Rule estimate (\(S_6\))**: **13** These are rounded as needed and based on the values visually estimated from the graph.