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 \(y=f(x) \geq 0\), 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