Let's break down the problem step by step: We are asked to estimate the definite integral \[ \int_{5}^{7} 3 \, dx \] using the **Midpoint Rule** with \( n = 8 \) subintervals and round the answer to three decimal places. --- ### **Step 1: Find the width of each subinterval (\(\Delta x\))** \[ \Delta x = \frac{b - a}{n} = \frac{7 - 5}{8} = \frac{2}{8} = 0.25 \] --- ### **Step 2: Determine the midpoints** The interval is \([5, 7]\), and each subinterval has width \(0.25\). The left endpoints are: \( 5, 5.25, 5.5, 5.75, 6, 6.25, 6.5, 6.75 \) The midpoints are halfway through each subinterval: \[ \text{Midpoints: } 5.125, 5.375, 5.625, 5.875, 6.125, 6.375, 6.625, 6.875 \] --- ### **Step 3: Apply the Midpoint Rule formula** The Midpoint Rule formula is: \[ M_n = \Delta x \left[ f(m_1) + f(m_2) + \cdots + f(m_n) \right] \] where \( m_i \) are the midpoints. But \( f(x) = 3 \) (a constant function), so \( f(m_i) = 3 \) for all \( i \). --- ### **Step 4: Substitute values** \[ M_8 = 0.25 \times \left[3 + 3 + 3 + 3 + 3 + 3 + 3 + 3\right] \] There are 8 terms of 3, so: \[ M_8 = 0.25 \times (8 \times 3) = 0.25 \times 24 = 6 \] --- ### **Step 5: Final Answer (to three decimal places)** \[ \boxed{M_8 = 6.000} \] --- #### **Explanation** Since the function is constant, the approximation is exact regardless of the rule or number of subintervals. The definite integral of \(3\) from \(5\) to \(7\) is exactly \(3 \times (7 - 5) = 6\).

Let's carefully analyze and solve the corrected problem: --- ### **Given:** Find the value of the definite integral: \[ \int_{5}^{7} \frac{3}{x} \, dx \] --- ### **Step 1: Recall the integral formula** The indefinite integral of \( \frac{1}{x} \) is: \[ \int \frac{1}{x} \, dx = \ln |x| + C \] Thus, \[ \int \frac{3}{x} \, dx = 3 \ln |x| + C \] --- ### **Step 2: Apply the limits** Using the Fundamental Theorem of Calculus: \[ \int_{5}^{7} \frac{3}{x} \, dx = 3 \left[ \ln |x| \right]_5^7 = 3 \left( \ln 7 - \ln 5 \right) \] --- ### **Step 3: Simplify** \[ = 3 \left( \ln \frac{7}{5} \right) \] --- ### **Step 4: Calculate \( \ln \frac{7}{5} \)** Using a calculator: \[ \frac{7}{5} = 1.4 \] \[ \ln 1.4 \approx 0.336472 \] Now multiply by 3: \[ 3 \times 0.336472 \approx 1.009416 \] --- ### **Final answer (rounded to 3 decimal places):** \[ \boxed{1.009} \] --- ### **Summary:** - The integral \( \int_{5}^{7} \frac{3}{x} \, dx \) evaluates to approximately **1.009** when rounded to three decimal places. --- **Note:** Always verify the integrand and limits carefully!