(Solved): Use the Midpoint Rule with \( n=5 \) to approximate the following integral. \[ \int_{1}^{2} \frac{ ...

Use the Midpoint Rule with \( n=5 \) to approximate the following integral. \[ \int_{1}^{2} \frac{3}{x} d x \] Solution The endpoints of the subintervals are \( 1, \frac{6}{5}, \frac{7}{5}, \frac{8}{5}, \frac{9}{5} \), and 2 , so the midpoints are \( 1.1,1.3,1.5,1.7 \), and . (See the figure below.) The width of the subintervals is \( \Delta x=\frac{(2-1)}{5}= \), so the Midpoint Rule gives the following. \[ \int_{1}^{2} \frac{3}{x} d x \approx \Delta x[f(1.1)+f(1.3)+f(1.5)+f(1.7)+f(1.9)] \] \[ =\frac{1}{5}\left(\frac{3}{1.1}+\frac{3}{1.3}+\frac{3}{1.5}+\frac{3}{1.7}+\frac{3}{1.9}\right) \] (1) \( \approx \quad \). (Round your answer to four decimal places.) Since \( f(x)=\frac{3}{x}>0 \) for \( 1 \leq x \leq 2 \), the integral represents an area, and the approximation given by the Midpoint Rule is the sum of the areas of the rectangles shown in the figure below.