Concepts and Vocabulary

### 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 $$A=\lim\limits_{n\rightarrow \infty }S_{n}$$.

9

Skill Building

### 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 $$A=4$$.

In Problems 7 and 8, refer to the illustrations. Approximate the shaded area under the graph of $$f$$ from 1 to 3:

1. By constructing rectangles using the left endpoint of each subinterval.
2. By constructing rectangles using the right endpoint of each subinterval.

### Question 5.7 ### Question 5.8 In Problems 9-12, partition each interval into $$n$$ subintervals each of the same length.

### 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 $$n = 16$$

In Problems 13 and 14, refer to the graphs. Approximate the shaded area:

1. By using lower sums $$s_{n}$$ (rectangles that lie below the graph of $$f$$).
2. By using upper sums $$S_{n}$$ (rectangles that lie above the graph of $$f$$).

### Question 5.13 ### Question 5.14 