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5.1 Assess Your Understanding

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Concepts and Vocabulary

  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.

Answers will vary.

  1. 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.

False

  1. If the closed interval [−2,4] is partitioned into 12 subintervals, each of the same length, then the length of each subinterval is ______.

12

  1. 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 Sn≥A and A=lim

True

351

Skill Building

  1. 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. (a) 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. (b) 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. (c) Compare the answers from parts (a) and (b) to the exact area A=9 and to the estimates obtained in Example 1.

  1. (a) Area is \dfrac{35}{4}.
  2. (b) Area is \dfrac{37}{4}.
  3. (c) \dfrac{35}{4}<9<\dfrac{37}{4}

  1. 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. (a) 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. (b) 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. (c) Compare the answers from parts (a) and (b) to the exact area A=4.

In Problems 7 and 8, refer to the graphs below. Approximate the shaded area under the graph of f:

  1. (a) By constructing rectangles using the left endpoint of each subinterval.
  2. (b) By constructing rectangles using the right endpoint of each subinterval.

  1. (a) 3
  2. (b) 6

In Problems 9–12, partition each interval into n subintervals each of the same length.

  1. [1,4] with n = 3

[1,2], [2,3], [3,4]

  1. [0,9] with n = 9

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

\left[-1,-\dfrac{1}{2}\right], \left[-\dfrac{1}{2},0\right], \left[0,\dfrac{1}{2}\right], \left[\dfrac{1}{2},1\right], \left[1,\dfrac{3}{2}\right], \left[\dfrac{3}{2},2\right], \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. [{-}4,4] with n = 16

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

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

  1. (a) 14
  2. (b) 48

  1. Area Under a Graph  Consider the area under the graph of y=x from 0 to 3.

    1. (a) Sketch the graph and the area under the graph.
    2. (b) Partition the interval [0,3] into n subintervals each of equal length.
    3. (c) Show that s_{n}=\sum\limits_{i=1}^{n}(i-1)\left(\dfrac{3}{n}\right)^{2}.
    4. (d) Show that S_{n}=\sum\limits_{i=1}^{n}{i\left(\dfrac{3}{n}\right)}^{2}.
    5. (e) Show that \lim\limits_{n\rightarrow \infty}s_{n}=\lim\limits_{n\rightarrow \infty } S_{n}={\dfrac{{9}}{{2}}}.

  1. (a)
  2. (b) \left[0,\dfrac{3}{n}\right], \left[\dfrac{3}{n},2\cdot\dfrac{3}{n}\right],\ldots, \left[(n-1)\cdot\dfrac{3}{n},3\right]
  3. (c) See Student Solutions Manual.
  4. (d) See Student Solutions Manual.
  5. (e) See Student Solutions Manual.

  1. Area Under a Graph  Consider the area under the graph of y=4x from 0 to 5.

    1. (a) Sketch the graph and the corresponding area.
    2. (b) Partition the interval [0,5] into n subintervals each of equal length.
    3. (c) Show that s_{n}=\sum\limits_{i=1}^{n}(i-1){\dfrac{100}{n^{2}}}.
    4. (d) Show that S_{n}=\sum\limits_{i=1}^{n} i{\dfrac{100}{n^{2}}}.
    5. (e) Show that \lim\limits_{n\rightarrow \infty} s_{n}={\lim\limits_{n\rightarrow \infty }}S_{n}=50.

In Problems 17–22, approximate the area A under the graph of each function f from a to b for n = 4 and n = 8 subintervals:

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

  1. f(x) = -x+10 on [0,8]

  1. (a) s_4=40, s_8=44
  2. (b) S_4=56, S_8=52

  1. f(x) = 2x + 5 on [2, 6]

  1. f(x) = 16-x^{2} on [0,4]

  1. (a) s_4=34, s_8=\dfrac{77}{2}
  2. (b) S_4=50, S_8=\dfrac{93}{2}

  1. f(x) = x^{3} on [0,8]

352

  1. f(x) = \cos x on \left[ -\dfrac{\pi }{2},\dfrac{\pi }{2}\right]

  1. (a) s_4=\dfrac{\sqrt{2}}{4}\pi\approx 1.111, s_8\approx 1.582
  2. (b) S_4=\dfrac{\sqrt{2}+2}{4}\pi\approx 2.682, S_8\approx 2.367

  1. f(x) = \sin x on [0,\pi]

  1. Rework Example 3 by using lower sums s_{n} (rectangles that lie below the graph of f).

s_n=\sum\limits_{i=1}^n \left(3(i-1)\dfrac{10}{n}\right)\dfrac{10}{n}=150-\dfrac{150}{n}; \lim\limits_{n\to \infty} s_n=150

  1. Rework Example 4 by using upper sums S_{n} (rectangles that lie above the graph of f).

In Problems 25–32, find the area A under the graph of f from a to b:

  1. (a) By using lower sums s_{n} (rectangles that lie below the graph of f).
  2. (b) By using upper sums S_{n}(rectangles that lie above the graph of f).
  3. (c) Compare the work required in (a) and (b). Which is easier? Could you have predicted this?

  1. f(x) =2x+1 from a = 0 to b = 4

  1. (a) A = \lim\limits_{n\to \infty} s_n =\lim\limits_{n\to \infty} \left(20- \dfrac{16}{n}\right)=20
  2. (b) A = \lim\limits_{n\to \infty} S_n =\lim\limits_{n\to \infty} \left(20 + \dfrac{16}{n}\right)=20
  3. (c) Answers will vary.

  1. f(x) = 3x+1 from a=0 to b=4

  1. f(x) =12-3x from a = 0 to b = 4

  1. (a) A = \lim\limits_{n\to \infty} s_n =\lim\limits_{n\to \infty} \left(24-\dfrac{24}{n}\right)=24
  2. (b) A = \lim\limits_{n\to \infty} S_n =\lim\limits_{n\to \infty} \left(24 + \dfrac{24}{n}\right)=24
  3. (c) Answers will vary.

  1. f(x) =5-x from a = 0 to b=4

  1. f(x) = 4 x^{2} from a = 0 to b = 2

  1. (a) A = \lim\limits_{n\to \infty} s_n =\lim\limits_{n\to \infty} \left(\dfrac{32}{3}-\dfrac{16}{n}+\dfrac{16}{3n^2}\right)=\dfrac{32}{3}
  2. (b) A = \lim\limits_{n\to \infty} S_n =\lim\limits_{n\to \infty} \left(\dfrac{32}{3}+\dfrac{16}{n}+\dfrac{16}{3n^2}\right)=\dfrac{32}{3}
  3. (c) Answers will vary.

  1. f(x) = \dfrac{1}{2}x^{2} from a = 0 to b=3

  1. f(x) = 4 - x^{2} from a = 0 to b = 2

  1. (a) A = \lim\limits_{n\to \infty} s_n =\lim\limits_{n\to \infty} \left(\dfrac{16}{3}-\dfrac{4}{n}-\dfrac{4}{3n^2}\right)=\dfrac{16}{3}
  2. (b) A = \lim\limits_{n\to \infty} S_n =\lim\limits_{n\to \infty} \left(\dfrac{16}{3}+\dfrac{4}{n}-\dfrac{4}{3n^2}\right)=\dfrac{16}{3}
  3. (c) Answers will vary.

  1. f(x) = 12-x^{2} from a = 0 to b = 3

Applications and Extensions

In Problems 33–38, find the area under the graph of f from a to b. [Hint: Partition the closed interval [a,b] into n subintervals [x_{0},x_{1}],[x_{1},x_{2}], \ldots , [x_{i-1},x_{i}], \ldots , [x_{n-1},x_{n}], where a=x_{0}\lt x_{1}\lt\cdots \lt x_{i}\lt\cdots \lt x_{n-1}\lt x_{n}=b, and each subinterval is of length \Delta x=\dfrac{b-a}{n}. As the figure below illustrates, the endpoints of each subinterval, written in terms of n, are \begin{eqnarray*} x_{0} &=& a, x_{1}=a+\dfrac{b-a}{n},\enspace x_{2}=a+2\left(\dfrac{b-a}{n}\right), \ldots , \\ x_{i-1} &=& a+(i-1) \left(\dfrac{b-a}{n} \right),\enspace x_{i}=a+i\left(\dfrac{b-a}{n}\right), \ldots ,\\ x_{n} &=& a+n \left(\dfrac{b-a}{n}\right) \end{eqnarray*}

  1. f(x)=x+3 from a = 1 to b=3

10

  1. f(x)=3-x from a=1 to b=3

  1. f(x)=2x+5 from a=-1 to b=2

18

  1. f(x)=2-3x from a=-2 to b=0

  1. f(x) = 2x^{2}+1 from a = 1 to b = 3

\dfrac{58}{3}

  1. f(x) = 4-x^{2} from a = 1 to b = 2

In Problems 39–42, approximate the area A under the graph of each function f by partitioning [a,b] into 20 subintervals of equal length and using an upper sum.

  1. f(x)=xe^{x} on [0,8]

A\approx 25{,}994

  1. f(x)=\ln x on [1,3]

  1. f(x)=\dfrac{1}{x} on [1,5]

A\approx 1.693

  1. f(x)=\dfrac{1}{x^{2}} on [2,6]

    1. (a) Graph y=\dfrac{4}{x} from x=1 to x=4 and shade the area under its graph.
    2. (b) Partition the interval [1,4] into n subintervals of equal length.
    3. (c) Show that the lower sum s_{n} is s_{n}=\sum\limits_{i=1}^{n}\dfrac{4}{\left( 1+\dfrac{3i}{n}\right) }\left(\dfrac{3}{n}\right).
    4. (d) Show that the upper sum S_{n} is S_{n}=\sum\limits_{i=1}^{n}\frac{4 }{\left( 1+\frac{3(i-1) }{n}\right) }\left( \frac{3}{n} \right)
    5. (e) Complete the following table:
      n 5 10 50 100
      s_{n}
      S_{n}
    6. (f) Use the table to give an upper and lower bound for the area.

  1. (a)
  2. (b) \left[1,1+\dfrac{3}{n}\right],\left[1+\dfrac{3}{n},1+2\cdot\dfrac{3}{n}\right],\ldots, \left[1+(n-1)\cdot\dfrac{3}{n},4\right].
  3. (c) See Student Solutions Manual.
  4. (d) See Student Solutions Manual.
  5. (e)
    n 5 10 50 100
    s_n 4.754 5.123 5.456 5.500
    S_n 6.554 6.023 5.636 5.590
  6. (f) 5.500\le A \le 5.590

Challenge Problems

  1. Area Under a Graph  Approximate the area under the graph of f(x)=x from a\geq 0 to b by using lower sums s_{n} and upper sums S_{n} for a partition of [a,b] into n subintervals, each of length \dfrac{b-a}{n}. Show that s_{n}\lt \frac{b^{2}-a^{2}}{2}\lt S_{n}

  1. Area Under a Graph  Approximate the area under the graph of f(x)=x^{2} from a\geq 0 to b by using lower sums s_{n} and upper sums S_{n} for a partition of [a,b] into n subintervals, each of length \dfrac{b-a}{n}. Show that s_{n}\lt \frac{b^{3}-a^{3}}{3}\lt S_{n}

See Student Solutions Manual.

  1. Area of a Right Triangle  Use lower sums s_{n} (rectangles that lie inside the triangle) and upper sums S_{n} (rectangles that lie outside the triangle) to find the area of a right triangle of height H and base B.

  1. Area of a Trapezoid  Use lower sums s_{n} (rectangles that lie inside the trapezoid) and upper sums S_{n} (rectangles that lie outside the trapezoid) to find the area of a trapezoid of heights H_1 and H_2 and base B.

See Student Solutions Manual.