Question 5.1

Explain how rectangles can be used to approximate the area enclosed by the graph of a function $y=f(x) \geq 0$, the $x$-axis, and the lines $x=a$ and $x=b$.

Question 5.2

True or False When a closed interval $[a,b]$ is partitioned into $n$ subintervals each of the same length, the length of each subinterval is $\dfrac{a+b}{n}$.

Question 5.3

If the closed interval $[-2,4]$ is partitioned into $12$ subintervals, each of the same length, then the length of each subinterval is _________.

Question 5.4

True or False If the area $A$ under the graph of a function $f$ that is continuous and nonnegative on a closed interval $[a,b]$ is approximated using upper sums $S_{n}$, then $S_{n}\geq A$ and $A=\lim\limits_{n\rightarrow \infty }S_{n}$.

Question 5.5

Approximate the area $A$ enclosed by the graph of $f(x) = \dfrac{1}{2}x+3$, the $x$-axis, and the lines $x=2$ and $x=4$ by partitioning the closed interval $[2, 4]$ into four subintervals: $$\left[ 2, \dfrac{5}{2}\right], \left[ \dfrac{5}{2},3 \right], \left[3, \dfrac{7}{2}\right], \left[ \dfrac{7}{2}, 4\right].$$

1. Using the left endpoint of each subinterval, draw four small rectangles that lie below the graph of $f$ and sum the areas of the four rectangles.
2. Using the right endpoint of each subinterval, draw four small rectangles that lie above the graph of $f$ and sum the areas of the four rectangles.
3. Compare the answers from parts (a) and (b) to the exact area $A=9$ and to the estimates obtained in Example 1.

Question 5.6

Approximate the area $A$ enclosed by the graph of $f(x) =6-2x$, the $x$-axis, and the lines $x=1$ and $x=3$ by partitioning the closed interval $[ 1,3]$ into four subintervals: $$\left[1, \dfrac{3}{2}\right], \left[ \dfrac{3}{2},2\right], \left[ 2,\dfrac{ 5}{2}\right], \left[ \dfrac{5}{2},3\right].$$

1. Using the right endpoint of each subinterval, draw four small rectangles that lie below the graph of $f$ and sum the areas of the four rectangles.
2. Using the left endpoint of each subinterval, draw four small rectangles that lie above the graph of $f$ and sum the areas of the four rectangles.
3. Compare the answers from parts (a) and (b) to the exact area $A=4$.

Question 5.7

Question 5.8

Question 5.9

$[1,4]$ with $n = 3$

Question 5.10

$[0,9]$ with $n = 9$

Question 5.11

$[{-}1,4]$ with $n = 10$

Question 5.12

$[{-}4,4]$ with $n = 16$

Question 5.13

Question 5.14