Answer
(a) $\frac{77}{60}$, underestimate.
(b) $\frac{25}{12}$, overestimate.
Work Step by Step
(a) Step 1. Identify the x-coordinates of the rectangles (right end points): range $[1,5]$, number of rectangles=$4$, x-coordinates $x_i=2,3,4,5$
Step 2. Use the given function $f(x)=\frac{1}{x}$ to calculate the heights of the rectangles as $f(x_i)=\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$
Step 3. Calculate the area by taking the sum of all the rectangle areas:
$$A=\sum^4_{i=1}1\times f(x_i)=\frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}=\frac{77}{60}$$
Where $1$ is the width of the rectangles, and the answer is $A=\frac{77}{60}$
Step 4. See graph, clearly it is an underestimate.
(b) For left end points, $x_i=1, 2,3,4$ and $f(x_i)=1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}$. The area can be estimated as
$A'=1+\frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}=\frac{25}{12}$. This time as shown in the graph, it is an overestimate.
