To estimate the area under the graph of $f(x) = \frac{5}{x}$ between $x = 1$ and $x = 17$ using the midpoint rule, we will calculate it with:

1. Two rectangles.
2. Four rectangles.

Step 1: Using Two Rectangles

• The interval length is $17 - 1 = 16$.
• Divide the interval into two subintervals: $[1, 9]$ and $[9, 17]$.
• Width of each rectangle: $\Delta x = \frac{16}{2} = 8$.
• Midpoints of the subintervals:
  • For $[1, 9]$: Midpoint = $\frac{1 + 9}{2} = 5$.
  • For $[9, 17]$: Midpoint = $\frac{9 + 17}{2} = 13$.

Now calculate the height of the rectangles using $f(x)$ at these midpoints:

• $f(5) = \frac{5}{5} = 1$.
• $f(13) = \frac{5}{13} \approx 0.3846$.

Area estimate: $\text{Area} = \Delta x \cdot \left[ f(5) + f(13) \right] = 8 \cdot (1 + 0.3846) \approx 8 \cdot 1.3846 = 11.0768.$

Step 2: Using Four Rectangles

• Divide the interval into four subintervals: $[1, 5]$, $[5, 9]$, $[9, 13]$, $[13, 17]$.
• Width of each rectangle: $\Delta x = \frac{16}{4} = 4$.
• Midpoints of the subintervals:
  • For $[1, 5]$: Midpoint = $\frac{1 + 5}{2} = 3$.
  • For $[5, 9]$: Midpoint = $\frac{5 + 9}{2} = 7$.
  • For $[9, 13]$: Midpoint = $\frac{9 + 13}{2} = 11$.
  • For $[13, 17]$: Midpoint = $\frac{13 + 17}{2} = 15$.

Now calculate the height of the rectangles using $f(x)$ at these midpoints:

• $f(3) = \frac{5}{3} \approx 1.6667$.
• $f(7) = \frac{5}{7} \approx 0.7143$.
• $f(11) = \frac{5}{11} \approx 0.4545$.
• $f(15) = \frac{5}{15} \approx 0.3333$.

Area estimate: