Answer
0.635 (3 S.F)
Work Step by Step
The question asks you to find the area under the curve using four approximating rectangles between x=1 and x=2. We can generally call these b and a (where 2 is b and 1 is a).
To find out the width of each rectangle, find the difference between the two x-values and divide by the number of rectangles (n):
$\frac{b-a}{n}$ = $\frac{2-1}{4}$ = 0.25
This means that the width of each rectangle will be 0.25. Therefore, the four rectangles will be situated between:
Rectangle 1: x = 1 and x = 1.25
Rectangle 2: x = 1.25 and x = 1.5
Rectangle 3: x = 1.5 and x = 1.75
Rectangle 4: x = 1.75 and x = 2
Next, you need to find the height or length of each rectangle. There are two ways to do this: using left endpoints (overestimate) or right endpoints (underestimate). Let's first use right endpoints.
For the graph of f(x) = $\frac{1}{x}$, find the values of y when x = 1.25, x = 1.5, x = 1.75 and x = 2 (right endpoints) using the graph.
When:
x = 1.25, y = 0.8
x = 1.5, y ≈ 0.667
x = 1.75, y ≈ 0.571
x = 2, y = 0.5
Now, you have to calculate the are of each rectangle.
Rectangle 1: 0.8 $\times$ 0.25 = 0.2
Rectangle 2: 0.6667 $\times$ 0.25 ≈ 0.167
Rectangle 3: 0.5714 $\times$ 0.25 ≈ 0.143
Rectangle 4: 0.5 $\times$ 0.25 = 0.125
Sum the values together to find the area:
0.2 + 0.167 + 0.143 + 0.125 = 0.635
The approximate area under the curve $\frac{1}{x}$ between x =1 and x = 2 using right endpoints is 0.635 (3 S.F).
