Calculus for Business, Economics, Life Sciences, and Social Sciences

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Chapter 6

Additional Integration Topics - all with Video Answers

Educators

Section 1

Area Between Curves

Problem 1

Use geometric formulas to find the area between the graphs of $y=f(x)$ and $y=g(x)$ over the indicated interval. (If necessary, review Appendix $\mathrm{C}$ ).
$$
f(x)=60, g(x)=45 ;[2,12]
$$

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

Use geometric formulas to find the area between the graphs of $y=f(x)$ and $y=g(x)$ over the indicated interval. (If necessary, review Appendix $\mathrm{C}$ ).
$$
f(x)=-30, g(x)=20 ;[-3,6]
$$

Narayan Hari

Numerade Educator

Problem 3

Use geometric formulas to find the area between the graphs of $y=f(x)$ and $y=g(x)$ over the indicated interval. (If necessary, review Appendix $\mathrm{C}$ ).
$$
f(x)=6+2 x, g(x)=6-x ;[0,5]
$$

Narayan Hari

Numerade Educator

Problem 4

Use geometric formulas to find the area between the graphs of $y=f(x)$ and $y=g(x)$ over the indicated interval. (If necessary, review Appendix $\mathrm{C}$ ).
$$
f(x)=0.5 x, g(x)=0.5 x-4 ;[0,8]
$$

Narayan Hari

Numerade Educator

Problem 5

Use geometric formulas to find the area between the graphs of $y=f(x)$ and $y=g(x)$ over the indicated interval. (If necessary, review Appendix $\mathrm{C}$ ).
$$
f(x)=-3-x, g(x)=4+2 x ;[-1,2]
$$

Narayan Hari

Numerade Educator

Problem 6

Use geometric formulas to find the area between the graphs of $y=f(x)$ and $y=g(x)$ over the indicated interval. (If necessary, review Appendix $\mathrm{C}$ ).
$$
f(x)=100-2 x, g(x)=10+3 x ;[5,10]
$$

Narayan Hari

Numerade Educator

Problem 7

Use geometric formulas to find the area between the graphs of $y=f(x)$ and $y=g(x)$ over the indicated interval. (If necessary, review Appendix $\mathrm{C}$ ).
$$
f(x)=x, g(x)=\sqrt{4-x^{2}} ;[0, \sqrt{2}]
$$

Narayan Hari

Numerade Educator

Problem 8

Use geometric formulas to find the area between the graphs of $y=f(x)$ and $y=g(x)$ over the indicated interval. (If necessary, review Appendix $\mathrm{C}$ ).
$$
f(x)=\sqrt{16-x^{2}}, g(x)=|x| ;[-2 \sqrt{2}, 2 \sqrt{2}]
$$

Bobby Barnes

University of North Texas

Problem 9

Refer to Figures A-D. Set up definite integrals in Problems 9-12 that represent the indicated shaded area.
Shaded area in Figure $\mathrm{B}$

Bobby Barnes

University of North Texas

Problem 10

Refer to Figures A-D. Set up definite integrals in Problems 9-12 that represent the indicated shaded area.
Shaded area in Figure A

Bobby Barnes

University of North Texas

Problem 11

Refer to Figures A-D. Set up definite integrals in Problems 9-12 that represent the indicated shaded area.
Shaded area in Figure $\mathrm{C}$

Bobby Barnes

University of North Texas

Problem 12

Refer to Figures A-D. Set up definite integrals in Problems 9-12 that represent the indicated shaded area.
Shaded area in Figure $\mathrm{D}$

Bobby Barnes

University of North Texas

Problem 13

Refer to Figures A-D. Set up definite integrals in Problems 9-12 that represent the indicated shaded area.
Explain why $\int_{a}^{b} h(x) d x$ does not represent the area between the graph of $y=h(x)$ and the $x$ axis from $x=a$ to $x=b$ in Figure $\mathrm{C}$.

Bobby Barnes

University of North Texas

Problem 14

Refer to Figures A-D. Set up definite integrals in Problems 9-12 that represent the indicated shaded area.
Explain why $\int_{a}^{b}[-h(x)] d x$ represents the area between the graph of $y=h(x)$ and the $x$ axis from $x=a$ to $x=b$ in Figure $\mathrm{C}$.

Bobby Barnes

University of North Texas

Problem 15

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=x+4 ; y=0 ; 0 \leq x \leq 4
$$

Narayan Hari

Numerade Educator

Problem 16

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-x+10 ; y=0 ;-2 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 17

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=x^{2}-20 ; y=0 ;-3 \leq x \leq 0
$$

Narayan Hari

Numerade Educator

Problem 18

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=x^{2}+2 ; y=0 ; 0 \leq x \leq 3
$$

Narayan Hari

Numerade Educator

Problem 19

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-x^{2}+10 ; y=0 ;-3 \leq x \leq 3
$$

Narayan Hari

Numerade Educator

Problem 20

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-2 x^{2} ; y=0 ;-6 \leq x \leq 0
$$

Narayan Hari

Numerade Educator

Problem 21

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=x^{3}+1 ; y=0 ; 0 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 22

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-x^{3}+3 ; y=0 ;-2 \leq x \leq 1
$$

Narayan Hari

Numerade Educator

Problem 23

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=x(1-x) ; y=0 ;-1 \leq x \leq 0
$$

Narayan Hari

Numerade Educator

Problem 24

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-x(3-x) ; y=0 ; 1 \leq x \leq 2
$$

Bobby Barnes

University of North Texas

Problem 25

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-e^{x} ; y=0 ;-1 \leq x \leq 1
$$

Bobby Barnes

University of North Texas

Problem 26

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=e^{x} ; y=0 ; 0 \leq x \leq 1
$$

Narayan Hari

Numerade Educator

Problem 27

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=\frac{1}{x} ; y=0 ; 1 \leq x \leq e
$$

Narayan Hari

Numerade Educator

Problem 28

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-\frac{1}{x} ; y=0 ;-1 \leq x \leq-\frac{1}{e}
$$

Narayan Hari

Numerade Educator

Problem 29

Base your answers on the Gini index of income concentration (see Table 2, page 388).
In which of Canada, Mexico, or the United States is income most equally distributed? Most unequally distributed?

Bobby Barnes

University of North Texas

Problem 30

Base your answers on the Gini index of income concentration (see Table 2, page 388).
In which of France, Germany, or Sweden is income most equally distributed? Most unequally distributed?

Bobby Barnes

University of North Texas

Problem 31

Base your answers on the Gini index of income concentration (see Table 2, page 388).
In which of Brazil, India, or Jordan is income most equally distributed? Most unequally distributed?

Bobby Barnes

University of North Texas

Problem 32

Base your answers on the Gini index of income concentration (see Table 2, page 388).
In which of China, Japan, or Russia is income most equally distributed? Most unequally distributed?

Bobby Barnes

University of North Texas

Problem 33

Refer to Figures $A$ and $B$. Set up definite integrals in Problems $33-40$ that represent the indicated shaded areas over the given intervals.
Over interval $[a, b]$ in Figure $A$

Bobby Barnes

University of North Texas

Problem 34

Refer to Figures $A$ and $B$. Set up definite integrals in Problems $33-40$ that represent the indicated shaded areas over the given intervals.
Over interval $[c, d]$ in Figure $A$

Bobby Barnes

University of North Texas

Problem 35

Refer to Figures $A$ and $B$. Set up definite integrals in Problems $33-40$ that represent the indicated shaded areas over the given intervals.
Over interval $[b, d]$ in Figure $A$

Bobby Barnes

University of North Texas

Problem 36

Refer to Figures $A$ and $B$. Set up definite integrals in Problems $33-40$ that represent the indicated shaded areas over the given intervals.
Over interval $[a, c]$ in Figure A

Bobby Barnes

University of North Texas

Problem 37

Refer to Figures $A$ and $B$. Set up definite integrals in Problems $33-40$ that represent the indicated shaded areas over the given intervals.
Over interval $[c, d]$ in Figure $\mathrm{B}$

Bobby Barnes

University of North Texas

Problem 38

Refer to Figures $A$ and $B$. Set up definite integrals in Problems $33-40$ that represent the indicated shaded areas over the given intervals.
Over interval $[a, b]$ in Figure $B$

Bobby Barnes

University of North Texas

Problem 39

Refer to Figures $A$ and $B$. Set up definite integrals in Problems $33-40$ that represent the indicated shaded areas over the given intervals.
Over interval $[a, c]$ in Figure $B$

Bobby Barnes

University of North Texas

Problem 40

Refer to Figures $A$ and $B$. Set up definite integrals in Problems $33-40$ that represent the indicated shaded areas over the given intervals.
Over interval $[b, d]$ in Figure $B$

Bobby Barnes

University of North Texas

Problem 41

Refer to Figures $A$ and $B$. Set up definite integrals in Problems $33-40$ that represent the indicated shaded areas over the given intervals.
Referring to Figure $\mathrm{B}$, explain how you would use definite integrals and the functions $f$ and $g$ to find the area bounded by the two functions from $x=a$ to $x=d$.

Bobby Barnes

University of North Texas

Problem 42

Refer to Figures $A$ and $B$. Set up definite integrals in Problems $33-40$ that represent the indicated shaded areas over the given intervals.
Referring to Figure A, explain how you would use definite integrals to find the area between the graph of $y=f(x)$ and the $x$ axis from $x=a$ to $x=d$.

Bobby Barnes

University of North Texas

Problem 43

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=-x ; y=0 ;-2 \leq x \leq 1
$$

Bobby Barnes

University of North Texas

Problem 44

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=-x+1 ; y=0 ;-1 \leq x \leq 2
$$

Bobby Barnes

University of North Texas

Problem 45

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{2}-4 ; y=0 ; 0 \leq x \leq 3
$$

Bobby Barnes

University of North Texas

Problem 46

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=4-x^{2} ; y=0 ; 0 \leq x \leq 4
$$

Bobby Barnes

University of North Texas

Problem 47

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{2}-3 x ; y=0 ;-2 \leq x \leq 2
$$

Bobby Barnes

University of North Texas

Problem 48

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=-x^{2}-2 x ; y=0 ;-2 \leq x \leq 1
$$

Bobby Barnes

University of North Texas

Problem 49

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=-2 x+8 ; y=12 ;-1 \leq x \leq 2
$$

Bobby Barnes

University of North Texas

Problem 50

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=2 x+6 ; y=3 ;-1 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 51

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=2 x+6 ; y=3 ;-1 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 52

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{2} ; y=9
$$

Narayan Hari

Numerade Educator

Problem 53

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=4-x^{2} ; y=-5
$$

Bobby Barnes

University of North Texas

Problem 54

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{2}-1 ; y=3
$$

Bobby Barnes

University of North Texas

Problem 55

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{2}+1 ; y=2 x-2 ;-1 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 56

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{2}-1 ; y=x-2 ;-2 \leq x \leq 1
$$

Narayan Hari

Numerade Educator

Problem 57

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=e^{0.5 x} ; y=-\frac{1}{x} ; 1 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 58

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=\frac{1}{x} ; y=-e^{x} ; 0.5 \leq x \leq 1
$$

Bobby Barnes

University of North Texas

Problem 59

Set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places. [Hint: A circle of radius $r$, with center at the origin, has equation $x^{2}+y^{2}=r^{2}$ and area $\left.\pi r^{2}\right] .$
$$
y=\sqrt{9-x^{2} ;} y=0 ;-3 \leq x \leq 3
$$

Narayan Hari

Numerade Educator

Problem 60

Set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places. [Hint: A circle of radius $r$, with center at the origin, has equation $x^{2}+y^{2}=r^{2}$ and area $\left.\pi r^{2}\right] .$
$$
y=\sqrt{25-x^{2}} ; y=0 ;-5 \leq x \leq 5
$$

Narayan Hari

Numerade Educator

Problem 61

Set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places. [Hint: A circle of radius $r$, with center at the origin, has equation $x^{2}+y^{2}=r^{2}$ and area $\left.\pi r^{2}\right] .$
$$
y=-\sqrt{16-x^{2}} ; y=0 ; 0 \leq x \leq 4
$$

Narayan Hari

Numerade Educator

Problem 62

Set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places. [Hint: A circle of radius $r$, with center at the origin, has equation $x^{2}+y^{2}=r^{2}$ and area $\left.\pi r^{2}\right] .$
$$
y=-\sqrt{36-x^{2}} ; y=0 ;-6 \leq x \leq 0
$$

Narayan Hari

Numerade Educator

Problem 63

Set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places. [Hint: A circle of radius $r$, with center at the origin, has equation $x^{2}+y^{2}=r^{2}$ and area $\left.\pi r^{2}\right] .$
$$
y=-\sqrt{4-x^{2}} ; y=\sqrt{4-x^{2}} ;-2 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 64

Set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places. [Hint: A circle of radius $r$, with center at the origin, has equation $x^{2}+y^{2}=r^{2}$ and area $\left.\pi r^{2}\right] .$
$$
y=-\sqrt{100-x^{2}} ; y=\sqrt{100-x^{2}} ;-10 \leq x \leq 10
$$

Bobby Barnes

University of North Texas

Problem 65

Find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=e^{x} ; y=e^{-x} ; 0 \leq x \leq 4
$$

Bobby Barnes

University of North Texas

Problem 66

Find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=e^{x} ; y=-e^{-x} ; 1 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 67

Find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=x^{3} ; y=4 x
$$

Bobby Barnes

University of North Texas

Problem 67

Find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=x^{3} ; y=4 x
$$

Bobby Barnes

University of North Texas

Problem 68

Find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=x^{3}+1 ; y=x+1
$$

Bobby Barnes

University of North Texas

Problem 69

Find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=x^{3}-3 x^{2}-9 x+12 ; y=x+12
$$

Bobby Barnes

University of North Texas

Problem 70

Find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=x^{3}-6 x^{2}+9 x ; y=x
$$

Bobby Barnes

University of North Texas

Problem 71

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=x^{3}-x^{2}+2 ; y=-x^{3}+8 x-2
$$

Bobby Barnes

University of North Texas

Problem 73

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=e^{-x} ; y=3-2 x
$$

Bobby Barnes

University of North Texas

Problem 74

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=2-(x+1)^{2} ; y=e^{x+1}
$$

Bobby Barnes

University of North Texas

Problem 75

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=e^{x} ; y=5 x-x^{3}
$$

Bobby Barnes

University of North Texas

Problem 76

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=2-e^{x} ; y=x^{3}+3 x^{2}
$$

Bobby Barnes

University of North Texas

Problem 77

Use a numerical integration routine on a graphing calculator to find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=e^{-x} ; y=\sqrt{\ln x ;} 2 \leq x \leq 5
$$

Wendi Zhao

Numerade Educator

Problem 78

Use a numerical integration routine on a graphing calculator to find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=x^{2}+3 x+1 ; y=e^{e^{x}} ;-3 \leq x \leq 0
$$

Wendi Zhao

Numerade Educator

Problem 79

Use a numerical integration routine on a graphing calculator to find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=e^{x^{2}} ; y=x+2
$$

Wendi Zhao

Numerade Educator

Problem 80

Use a numerical integration routine on a graphing calculator to find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=\ln (\ln x) ; y=0.01 x
$$

Wendi Zhao

Numerade Educator

Problem 81

Find the constant $c$ ( to 2 decimal places) such that the Lorenz curve $f(x)=x^{c}$ has the given Gini index of income concentration.
$$
0.52
$$

Bobby Barnes

University of North Texas

Problem 82

Find the constant $c$ ( to 2 decimal places) such that the Lorenz curve $f(x)=x^{c}$ has the given Gini index of income concentration.
$$
0.23
$$

Bobby Barnes

University of North Texas

Problem 83

Find the constant $c$ ( to 2 decimal places) such that the Lorenz curve $f(x)=x^{c}$ has the given Gini index of income concentration.
$$
0.29
$$

Bobby Barnes

University of North Texas

Problem 84

Find the constant $c$ ( to 2 decimal places) such that the Lorenz curve $f(x)=x^{c}$ has the given Gini index of income concentration.
$$
0.65
$$

Bobby Barnes

University of North Texas

Problem 85

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
Using production and geological data, the management of an oil company estimates that oil will be pumped from a producing field at a rate given by
$$
R(t)=\frac{100}{t+10}+10 \quad 0 \leq t \leq 15
$$
where $R(t)$ is the rate of production (in thousands of barrels per year) $t$ years after pumping begins. Find the area between the graph of $R$ and the $t$ axis over the interval [5,10] and interpret the results.

Bobby Barnes

University of North Texas

Problem 86

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
In Problem $85,$ if the rate is found to be
$$
R(t)=\frac{100 t}{t^{2}+25}+4 \quad 0 \leq t \leq 25
$$
then find the area between the graph of $R$ and the $t$ axis over the interval [5,15] and interpret the results.

Bobby Barnes

University of North Texas

Problem 87

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
An amusement company maintains records for each video game it installs in an arcade. Suppose that $C(t)$ and $R(t)$ represent the total accumulated costs and revenues (in thousands of dollars), respectively, $t$ years after a particular game has been installed. If
$$
C^{\prime}(t)=2 \quad \text { and } \quad R^{\prime}(t)=9 e^{-0.3 t}
$$
then find the area between the graphs of $C^{\prime}$ and $R^{\prime}$ over the interval on the $t$ axis from 0 to the useful life of the game and interpret the results.

Wendi Zhao

Numerade Educator

Problem 88

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
Repeat Problem 87 if
$$
C^{\prime}(t)=2 t \quad \text { and } \quad R^{\prime}(t)=5 t e^{-0.1 t^{2}}
$$

Wendi Zhao

Numerade Educator

Problem 89

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
In a study on the effects of World War II on the U.S. economy, an economist used data from the
U.S. Census Bureau to produce the following Lorenz curves for the distribution of U.S. income in 1935 and in 1947:
$$
f(x)=x^{2.4} \quad \text { Lorenz curve for } 1935
$$
$$
g(x)=x^{1.6} \quad \text { Lorenz curve for } 1947
$$
Find the Gini index of income concentration for each Lorenz $z$ curve and interpret the results.

Bobby Barnes

University of North Texas

Problem 90

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
Using data from the U.S. Census Bureau, an economist produced the following Lorenz curves for the distribution of U.S. income in 1962 and in 1972 :
$$
f(x)=\frac{3}{10} x+\frac{7}{10} x^{2} \quad \text { Lorenz curve for } 1962
$$
$$
g(x)=\frac{1}{2} x+\frac{1}{2} x^{2} \quad \text { Lorenz curve for } 1972
$$
Find the Gini index of income concentration for each Lorenz
curve and interpret the results.

Wendi Zhao

Numerade Educator

Problem 91

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
Lorenz curves also can provide a relative measure of the distribution of a country's total assets. Using data in a report by the U.S. Congressional Joint Economic Committee, an economist produced the following Lorenz curves for the distribution of total U.S. assets in 1963 and in 1983:
$$
f(x)=x^{10} \quad \text { Lorenz curve for } 1963
$$
$$
g(x)=x^{12} \quad \text { Lorenz curve for } 1983
$$
Find the Gini index of income concentration for each Lorenz curve and interpret the results.

Bobby Barnes

University of North Texas

Problem 92

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
The government of a small country is planning sweeping changes in the tax structure in order to provide a more equitable distribution of income. The Lorenz curves for the current income distribution and for the projected income distribution after enactment of the tax changes are as follows:
$$
f(x)=x^{2.3} \quad \text { Current Lorenz curve Projected Lorenz curve after changes in tax laws }
$$
$$
g(x)=0.4 x+0.6 x^{2}
$$

Bobby Barnes

University of North Texas

Problem 93

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
The data in the following table describe the distribution of wealth in a country:
$$
\begin{array}{ccccccc}
x & 0 & 0.20 & 0.40 & 0.60 & 0.80 & 1 \\
\hline y & 0 & 0.12 & 0.31 & 0.54 & 0.78 & 1
\end{array}
$$
(A) Use quadratic regression to find the equation of a Lorenz curve for the data.
(B) Use the regression equation and a numerical integration routine to approximate the Gini index of income concentration.

Bobby Barnes

University of North Texas

Problem 94

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
Refer to Problem $93 .$
(A) Use cubic regression to find the equation of a Lorenz curve for the data.
(B) Use the cubic regression equation you found in Part (A) and a numerical integration routine to approximate the Gini index of income concentration.

Bobby Barnes

University of North Texas

Problem 95

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
A yeast culture is growing at a rate of $W^{\prime}(t)=$ $0.3 e^{0.1 t}$ grams per hour. Find the area between the graph of $W^{\prime}$ and the $t$ axis over the interval [0,10] and interpret the results.

Bobby Barnes

University of North Texas

Problem 96

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
The instantaneous rate of change in demand for U.S. lumber since $1970(t=0)$, in billions of cubic feet per year, is given by
$$
Q^{\prime}(t)=12+0.006 t^{2} \quad 0 \leq t \leq 50
$$
Find the area between the graph of $Q^{\prime}$ and the $t$ axis over the interval $[35,40],$ and interpret the results.

Bobby Barnes

University of North Texas

Problem 97

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
A college language class was chosen for a learning experiment. Using a list of 50 words, the experiment measured the rate of vocabulary memorization at different times during a continuous 5 -hour study session. The average rate of learning for the entire class was inversely proportional to the time spent studying and was given approximately by
$$
V^{\prime}(t)=\frac{15}{t} \quad 1 \leq t \leq 5
$$
Find the area between the graph of $V^{\prime}$ and the $t$ axis over the interval $[2,4],$ and interpret the results.

Narayan Hari

Numerade Educator

Problem 98

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
Repeat Problem 97 if $V^{\prime}(t)=13 / t^{1 / 2}$ and the interval is changed to [1,4] .

Narayan Hari

Numerade Educator