Question

Use four rectangles to estimate the area between the graph of the function f(x)=4/(7x)and the x-axis on the interval[1, 5] using the left endpoints of the subintervals as the sample points. Write the exact answer. Do not round.

Answer 1

We need to use the left endpoints as the length of each one of the four rectangles to estimate the area below the graph of

`f(x)=(4)/(7x)`

Since we need to divide the interval [1, 5] into four subintervals, the width of each rectangle will be 1:

Step 1

Find the height of the rectangles by evaluating f(x) at x = 1, 2, 3, and 4:

`\begin{gathered} f(1)=(4)/(7\cdot1)=(4)/(7) \\ \\ \cdot f(2)=(4)/(7\cdot2)=(4)/(14)=(2)/(7) \\ \\ f(3)=(4)/(7\cdot3)=(4)/(21) \\ \\ f(4)=(4)/(7\cdot4)=(1)/(7) \end{gathered}`

Step 2

Multiplying the height of each rectangle by its width (1), we find the areas of the rectangles. Since the widths are 1, the areas are numerically equal to the heights.

Then we add those areas to estimate the area between the curve and the x-axis in the interval [1, 5]:

`(4)/(7)+(2)/(7)+(4)/(21)+(1)/(7)=(7)/(7)+(4)/(21)=(21)/(21)+(4)/(21)=(25)/(21)`

Answer

Therefore

`\text{Left-endpoint est. }=(25)/(21)`