Answer
(a) $70$, underestimate.
(b) $95$, overestimate.
Work Step by Step
Step 1. Identify the x-coordinates of the rectangles (right endpoints): range $[0,5]$, number of rectangles=$5$, x-coordinates $x_i=1,2,3,4,5$
Step 2. Use the given function $f(x)=25-x^2$ to calculate the heights of the rectangles as $f(x_i)=24,21,16,9,0$.
Step 3. Calculate the area by taking the sum of all the rectangle areas:
$$A=\sum^5_{i=1}1f(x_i)=24+21+16+9+0=70$$
Where $1$ is the width of the rectangles, and the answer is $A=70$
Step 4. See graph, clearly it is an underestimate.
(b) Step 1. Identify the x-coordinates of the rectangles (left endpoints): range $[0,5]$, number of rectangles=$5$, x-coordinates $x_i=0, 1,2,3,4$
Step 2. Use the given function $f(x)=25-x^2$ to calculate the heights of the rectangles as $f(x_i)=25, 24,21,16,9$.
Step 3. Calculate the area by taking the sum of all the rectangle areas:
$$A'=\sum^5_{i=1}1f(x_i)=25+24+21+16+9=95$$
Where $1$ is the width of the rectangles, and the answer is $A'=95$
Step 4. See graph, clearly it is an overestimate.
