Chapter 13.5, Problem 4E

a.

To determine

To find the area of the graph of $f,x\in \left[1,5\right]$ .

Answer to Problem 4E

The area of the graph of $f$ is $86.6\text{ and }70.6$ .

Explanation of Solution

Calculation:

Sketch six rectangles by use an upper estimate for area under the curve and find the area by add the areas of each rectangle. This is left endpoint.

  

The area of add all the rectangles will be,

  $\begin{array}{l}{L}_6=2\left(9+8.8+8.2+7.3+5.9+4.1\right)\\ L_6\approx 86.6\end{array}$

ii) Sketch six rectangles by use a lower estimate for area under the curve and find the area by add the areas of each rectangle. Start from right endpoint.

  

  $\begin{array}{l}{R}_6=86.6+2\left(1\right)-2\left(9\right)\\ =70.6\end{array}$

Thus, the area of the graph of $f$ is $86.6\text{ and }70.6$ .

b.

To determine

To find the $L_6$ is an overestimate or underestimate.

Answer to Problem 4E

  $L_6$ is an overestimate by use left end points.

Explanation of Solution

Given: $L_6$ : to find the overestimate or underestimate.

Calculation:

From the above graphs,

Since, $L_6$ is an overestimate by use left endpoints.

Thus $L_6$ is an overestimate by use left endpoints.

c.

To determine

To find $R_6$ is an overestimate or underestimate.

Answer to Problem 4E

  $R_6$ is an underestimate by use right end points.

Explanation of Solution

Calculation:

From the above graphs,

Since, $R_6$ is an underestimate by use right endpoints.

Thus $R_6$ is an underestimate by use right endpoints.