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Calculus for AP

Jon Rogawski & Ray Cannon

Chapter 6

APPLICATIONS OF THE INTEGRAL

Educators


Problem 1

Find the area of the region between $y=3 x^{2}+12$ and $y=4 x+4$ over $[-3,3]$ (Figure 9).

Linh V.
Numerade Educator

Problem 2

Find the area of the region between the graphs of $f(x)=3 x+8$ and $g(x)=x^{2}+2 x+2$ over $[0,2]$

Amy J.
Numerade Educator

Problem 3

Find the area of theregionenclosed by the graphs of $f(x)=x^{2}+2$ and $g(x)=2 x+5($ Figure 10$) .$

Linh V.
Numerade Educator

Problem 5

In Exercises 5 and $6,$ sketch the region between $y=\sin x$ and $y=\cos x$ over the interval and find its area.$$
\left[\frac{\pi}{4}, \frac{\pi}{2}\right]
$$

Linh V.
Numerade Educator

Problem 6

In Exercises 5 and $6,$ sketch the region between $y=\sin x$ and $y=\cos x$ over the interval and find its area.
$$
[0, \pi]
$$

Amy J.
Numerade Educator

Problem 7

In Exercises 7 and 8 ,$ let $f(x)=20+x-x^{2}$ and $g(x)=x^{2}-5x
Sketch the region enclosed by the graphs of $f(x)$ and $g(x)$ and compute its area.

Linh V.
Numerade Educator

Problem 8

In Exercises 7 and 8 ,$ let $f(x)=20+x-x^{2}$ and $g(x)=x^{2}-5x
Sketch the region between the graphs of $f(x)$ and $g(x)$ over $[4,8]$ and compute its area as a sum of two integrals.

Amy J.
Numerade Educator

Problem 9

Find the area between $y=e^{x}$ and $y=e^{2 x}$ over $[0,1]$

Linh V.
Numerade Educator

Problem 10

Find the area of the region bounded by $y=e^{x}$ and $y=12-e^{x}$ and the $y$ -axis.

Amy J.
Numerade Educator

Problem 11

Sketch the region bounded by the line $y=2$ and the graph of $y=\sec ^{2} x$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$ and find its area.

Linh V.
Numerade Educator

Problem 12

Sketch the region bounded by
$$
y=\frac{1}{\sqrt{1-x^{2}}} \quad \text { and } \quad y=-\frac{1}{\sqrt{1-x^{2}}}
$$
for $-\frac{1}{2} \leq x \leq \frac{1}{2}$ and find its area.

Amy J.
Numerade Educator

Problem 13

In Exercises $13-16,$ find the area of the shaded region in Figures $12-15$

Linh V.
Numerade Educator

Problem 14

In Exercises $13-16,$ find the area of the shaded region in Figures $12-15$

Amy J.
Numerade Educator

Problem 15

In Exercises $13-16,$ find the area of the shaded region in Figures $12-15$

Linh V.
Numerade Educator

Problem 16

In Exercises $13-16,$ find the area of the shaded region in Figures $12-15$

Amy J.
Numerade Educator

Problem 17

In Exercises 17 and $18,$ find the area between the graphs of $x=\sin y$ and$x=1-\cos y$ over the given interval (Figure 16$) .$
$$
0 \leq y \leq \frac{\pi}{2}
$$

Linh V.
Numerade Educator

Problem 18

In Exercises 17 and $18,$ find the area between the graphs of $x=\sin y$ and $x=1-$ cos $y$ over the given interval (Figure 16$) .$
$$
-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}
$$

Amy J.
Numerade Educator

Problem 19

Find the area of the region lying to the right of $x=y^{2}+4 y-22_{i}$ . and to the left of $x=3 y+8 .$

Linh V.
Numerade Educator

Problem 20

Find the area of the region lying to the right of $x=y^{2}-5$ and to the left of $x=3-y^{2}$ .

Amy J.
Numerade Educator

Problem 21

Figure 17 shows the region enclosed by $x=y^{3}-26 y+10$ and $x=40-6 y^{2}-y^{3} .$ Match the equations with the curves and compute the area of the region.

Linh V.
Numerade Educator

Problem 22

Figure 18 shows the region enclosed by $y=x^{3}-6 x$ and $y=$ $8-3 x^{2} .$ Match the equations with the curves and compute the area of the region.

Amy J.
Numerade Educator

Problem 23

In Exercises 23 and $24,$ find the area enclosed by the graphs in two ways: by integrating along the $x$ -axis and by integrating along the $y$ -axis.
$$
x=9-y^{2}, \quad x=5
$$

Linh V.
Numerade Educator

Problem 24

In Exercises 23 and $24,$ find the area enclosed by the graphs in two ways: by integrating along the $x$ -axis and by integrating along the $y$ -axis.
$$
\text{The semicubical parabola} y^{2}=x^{3} \text { and the line } x=1
$$

Amy J.
Numerade Educator

Problem 25

In Exercises 25 and $26,$ find the area of the region using the method (integration along either the $x$ -or the $y$ -axis) that requires you to evaluate just one integral.
$$\text{Region between } y^{2}=x+5 \text{ and } y^{2}=3-x$$

Linh V.
Numerade Educator

Problem 26

In Exercises 25 and $26,$ find the area of the region using the method (integration along either the $x$ -or the $y$ -axis) that requires you to evaluate just one integral. $$\text{Region between } y=x \text{ and } x + y = 8 \text{ over } [2,3]$$

Amy J.
Numerade Educator

Problem 27

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
y=4-x^{2}, \quad y=x^{2}-4
$$

Linh V.
Numerade Educator

Problem 28

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
y=x^{2}-6, \quad y=6-x^{3}, \quad y \text{-axis}
$$

Amy J.
Numerade Educator

Problem 29

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
x+y=4, \quad x-y=0, \quad y+3 x=4
$$

Linh V.
Numerade Educator

Problem 30

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
y=8-3 x, \quad y=6-x, \quad y=2
$$

Amy J.
Numerade Educator

Problem 31

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
y=8-\sqrt{x}, \quad y=\sqrt{x}, \quad x=0
$$

Linh V.
Numerade Educator

Problem 32

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
y=\frac{x}{x^{2}+1}, \quad y=\frac{x}{5}
$$

Amy J.
Numerade Educator

Problem 33

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
x=|y|, \quad x=1-|y|
$$

Linh V.
Numerade Educator

Problem 34

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
y=|x|, \quad y=x^{2}-6
$$

Amy J.
Numerade Educator

Problem 35

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
x=y^{3}-18 y, \quad y+2 x=0
$$

Linh V.
Numerade Educator

Problem 36

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
y=x \sqrt{x-2}, \quad y=-x \sqrt{x-2}, \quad x=4
$$

Amy J.
Numerade Educator

Problem 37

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
x=2 y, \quad x+1=(y-1)^{2}
$$

Linh V.
Numerade Educator

Problem 38

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
x+y=1, \quad x^{1 / 2}+y^{1 / 2}=1
$$

Amy J.
Numerade Educator

Problem 39

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
y=\cos x, \quad y=\cos 2 x, \quad x=0, \quad x=\frac{2 \pi}{3}
$$

Linh V.
Numerade Educator

Problem 40

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
x=\tan x, \quad y=-\tan x, \quad x=\frac{\pi}{4}
$$

Amy J.
Numerade Educator

Problem 41

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
y=\sin x, \quad y=\csc ^{2} x, \quad x=\frac{\pi}{4}
$$

Linh V.
Numerade Educator

Problem 42

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
x=\sin y, \quad x=\frac{2}{\pi} y
$$

Amy J.
Numerade Educator

Problem 43

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
y=e^{x}, \quad y=e^{-x}, \quad y=2
$$

Linh V.
Numerade Educator

Problem 44

In Exercises $27-44,$ sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.
$$
y=\frac{\ln x}{x}, \quad y=\frac{(\ln x)^{2}}{x}
$$

Amy J.
Numerade Educator

Problem 45

Plot
$$
y=\frac{x}{\sqrt{x^{2}+1}} \quad \text { and } \quad y=(x-1)^{2}
$$
on the same set of axes. Use a computer algebra system to find the
points of intersection numerically and compute the area between the
curves.

Linh V.
Numerade Educator

Problem 46

Sketch a region whose area is represented by
$$
\int_{-\sqrt{2} / 2}^{\sqrt{2} / 2}\left(\sqrt{1-x^{2}}-|x|\right) d x
$$
and evaluate using geometry.

Amy J.
Numerade Educator

Problem 47

Athletes 1 and 2 run along a straight track with velocities $v_{1}(t)$ and $v_{2}(t)($ in $\mathrm{m} / \mathrm{s})$ as shown in Figure 19
$$
\begin{array}{l}{\text { (a) Which of the following is represented by the area of the shaded }} \\ {\text { region over }[0,10] \text { ? }} \\ {\text { i. The distance between athletes } 1 \text { and } 2 \text { at time } t=10 \mathrm{s} \text { . }} \\ {\text { ii. The difference in the distance traveled by the athletes over the time }} \\ {\text { interval }[0,10] .}\\{\text { (b) Does Figure } 19 \text { give us enough information to determine who is }} \\ {\text { ahead at time } t=10 \mathrm{s} ?} \\ {\text { (c) If the athletes begin at the same time and place, who is ahead at }} \\ {t=10 \text { s? At } t=25 \mathrm{s} ?}\end{array}
$$

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Problem 48

Express the area (not signed) of the shaded region in Figure 20 as a sum of three integrals involving $f(x)$ and $g(x)$ .

Amy J.
Numerade Educator

Problem 49

Find the area enclosed by the curves $y=c-x^{2}$ and $y=x^{2}-c$as a function of $c .$ Find the value of $c$ for which this area is equal to $1 .$

Linh V.
Numerade Educator

Problem 50

Set up (but do not evaluate) an integral that expresses the area between the circles $x^{2}+y^{2}=2$ and $x^{2}+(y-1)^{2}=1$

Amy J.
Numerade Educator

Problem 51

Set up (but do not evaluate) an integral that expresses the area between the graphs of $y=\left(1+x^{2}\right)^{-1}$ and $y=x^{2}$

Linh V.
Numerade Educator

Problem 52

Find a numerical approximation to the area above $y=1-(x / \pi)$ and below $y=\sin x$ (find the points of intersection numerically).

Amy J.
Numerade Educator

Problem 53

Find a numerical approximation to the area above $y=|x|$ and below $y=\cos x .$

Linh V.
Numerade Educator

Problem 54

Use a computer algebra system to find a numerical approximation to the number $c$ (besides zero) in $\left[0, \frac{\pi}{2}\right],$ where the curves $y=\sin x$ and $y=\tan ^{2} x$ intersect. Then find the area enclosed by the graphs over $[0, c] .$

Amy J.
Numerade Educator

Problem 55

The back of Jon's guitar (Figure 21$)$ is 19 inches long. Jon measured the width at 1 -in. intervals, beginning and ending $\frac{1}{2}$ in. from the ends, obtaining the results. Use the midpoint rule to estimate the area of the back.

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Problem 56

Referring to Figure 1 at the beginning of this section, estimate the projected number of additional joules produced in the years $2009-2030$ as a result of government stimulus spending in $2009-2010 .$ Note: One watt is equal to one joule per second, and one gigawatt is $10^{9}$ watts.

Amy J.
Numerade Educator

Problem 57

Exercises 57 and 58 use the notation and results of Exercises $49-51$ of Section $3.4 .$ For a given country, $F(r)$ is the fraction of total income that goes to the bottom rth fraction of households. The graph of $y=F(r)$ is called the Lorenz curve.
Let $A$ be the area between $y=r$ and $y=F(r)$ over the interval $[0,1]($ Figure 22$) .$ The Gini index is the ratio $G=A / B,$ where $B$ is the area under $y=r$ over $[0,1]$ .
$$
\begin{array}{l}{\text { (a) Show that } G=2 \int_{0}^{1}(r-F(r)) d r} \\ {\text { (b) Calculate } G \text { if }}\end{array}
$$
$$
F(r)=\left\{\begin{array}{ll}{\frac{1}{3} r} & {\text { for } 0 \leq r \leq \frac{1}{2}} \\ {\frac{5}{3} r-\frac{2}{3}} & {\text { for } \frac{1}{2} \leq r \leq 1}\end{array}\right.
$$
$$
\begin{array}{l}{\text { (c) The Gini index is a measure of income distribution, with a lower }} \\ {\text { value indicating a more equal distribution. Calculate } G \text { if } F(r)=r \text { (in }} \\ {\text { this case, all households have the same income by Exercise } 51(b) \text { of }} \\ {\text { Section } 3.4 \text { ). }} \\ {\text { (d) What is } G \text { if all of the income goes to one household? Hint: In this }} \\ {\text { extreme case, } F(r)=0 \text { for } 0 \leq r<1}\end{array}
$$

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Problem 58

Exercises 57 and 58 use the notation and results of Exercises $49-51$ of Section $3.4 .$ For a given country, $F(r)$ is the fraction of total income that goes to the bottom rth fraction of households. The graph of $y=F(r)$ is called the Lorenz curve.
Let $A$ be the area between $y=r$ and $y=F(r)$ over the interval $[0,1]($ Figure 22$) .$ The Gini index is the ratio $G=A / B,$ where $B$ is the area under $y=r$ over $[0,1]$ .
Calculate the Gini index of the United States in the year 2001 from the Lorenz curve in Figure $22,$ which consists of segments joining the data points in the following table.
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline r & {0} & {0.2} & {0.4} & {0.6} & {0.8} & {1} \\ \hline F(r) & {0} & {0.035} & {0.123} & {0.269} & {0.499} & {1} \\ \hline\end{array}
$$.

Amy J.
Numerade Educator

Problem 59

Find the line $y=m x$ that divides the area under the curve $y=$ $x(1-x)$ over $[0,1]$ into two regions of equal area.

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Problem 60

Let $c$ be the number such that the area under $y=\sin x$ over $[0, \pi]$ is divided in half by the line $y=c x$ (Figure 23$) .$ Find an equation for $c$ and solve this equation numerically using a computer
algebra system.

Amy J.
Numerade Educator

Problem 61

Explain geometrically (without calculation):
$$
\int_{0}^{1} x^{n} d x+\int_{0}^{1} x^{1 / n} d x=1 \quad \text { (for } n>0 )
$$

Linh V.
Numerade Educator

Problem 62

Let $f(x)$ be an increasing function with inverse $g(x) .$ Explain geometrically:
$$
\int_{0}^{a} f(x) d x+\int_{f(0)}^{f(a)} g(x) d x=a f(a)
$$

Amy J.
Numerade Educator