Answer
(a) $8$, overestimate.
(b) $5$, underestimate.
Work Step by Step
(a) Step 1. Identify the x-coordinates of the rectangles (right endpoints): range $[-1,2]$, number of rectangles=$3$, x-coordinates $x_i=0,1,2$
Step 2. Use the given function $f(x)=1+x^2$ to calculate the heights of the rectangles as $f(x_i)=1,2,5$.
Step 3. Calculate the area by taking the sum of all the rectangle areas:
$$A=\sum^3_{i=1}1f(x_i)=1+2+5=8$$
Where $1$ is the width of the rectangles, and the answer is $A=8$
Step 4. See graph, it is an overestimate.
(b) Step 1. Identify the x-coordinates of the rectangles (left endpoints): range $[-1,2]$, number of rectangles=$3$, x-coordinates $x_i=-1,0,1$
Step 2. Use the given function $f(x)=1+x^2$ to calculate the heights of the rectangles as $f(x_i)=2,1,2$.
Step 3. Calculate the area by taking the sum of all the rectangle areas:
$$A'=\sum^3_{i=1}1f(x_i)=2+1+2=5$$
Where $1$ is the width of the rectangles, and the answer is $A'=5$
Step 4. See graph, it is an underestimate.
