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5.2 Assess Your Understanding

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)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 a+bn.

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 Sn, then SnA and A=lim.

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