Answer
Two rectangles: $1.5$
Four rectangles: $1.5746031746$
Work Step by Step
Given-ƒ(x) = $\frac{1}{x}$ between x = 1 and x = 5, using the midpoint rule, we'll divide the interval [1, 5] into rectangles and calculate the areas of those rectangles.
First, let's use two rectangles:
Divide the interval [1, 5] into two equal subintervals: [1, 3] and [3, 5].
Calculate the midpoints of the bases of the rectangles:
For the first rectangle: midpoint = $\frac{(1 + 3) }{2}$= 2
For the second rectangle: midpoint = $\frac{ (3 + 5)}{2}$= 4
Evaluate the function ƒ(x) = $\frac{1}{x}$ at the midpoints:
For the first rectangle: ƒ(2) = $\frac{1}{2}$= 0.5
For the second rectangle: ƒ(4) = $\frac{1}{4}$= 0.25
Calculate the areas of the rectangles:
For the first rectangle: area = 2 $\times$ 0.5 = 1
For the second rectangle: area = 2 $\times$ 0.25 = 0.5
Sum up the areas of the rectangles: total area ≈ 1 + 0.5 = 1.5
Now, let's use four rectangles:
Divide the interval [1, 5] into four equal subintervals: [1, 2], [2, 3], [3, 4], and [4, 5].
Calculate the midpoints of the bases of the rectangles:
For the first rectangle: midpoint = $\frac{(1 + 2)}{2}$= 1.5
For the second rectangle: midpoint = $\frac{(2 + 3)}{2}$= 2.5
For the third rectangle: midpoint = $\frac{(3 + 4) }{2}$= 3.5
For the fourth rectangle: midpoint = $\frac{(4 + 5)}{2}$= 4.5
Evaluate the function ƒ(x) = $\frac{1}{x}$ at the midpoints:
For the first rectangle: ƒ(1.5) = $\frac{1}{1.5}$ = 0.66666666667
For the second rectangle: ƒ(2.5) = $\frac{1}{2.5}$ = 0.4
For the third rectangle: ƒ(3.5) = $\frac{1}{3.5}$= 0.28571428571
For the fourth rectangle: ƒ(4.5) = $\frac{1}{4.5}$= 0.22222222222
Calculate the areas of the rectangles:
For each rectangle: area = 1 $\times$ ƒ(midpoint)
For the first rectangle: area = 1 $\times$ 0.66666666667 = 0.66666666667
For the second rectangle: area = 1 $\times$ 0.4 = 0.4
For the third rectangle: area = 1 $\times$ 0.28571428571 = 0.28571428571
For the fourth rectangle: area = 1 $\times$ 0.22222222222 = 0.22222222222
Sum up the areas of the rectangles: total area ≈ 0.66666666667 + 0.4 + 0.28571428571 + 0.22222222222 ≈ 1.5746031746
