Calculus

Ron Larson, Robert P. Hostetler, Bruce H. Edwards

Chapter 7

Applications of Integration - all with Video Answers

Educators

Section 1

Area of a Region Between Two Curves

Problem 1

Set up the definite integral that gives the area of the region.
$$
\begin{aligned}
&f(x)=x^{2}-6 x \\
&g(x)=0
\end{aligned}
$$

Uma Kumari

Uma Kumari

Numerade Educator

Problem 2

Set up the definite integral that gives the area of the region.
$$
\begin{aligned}
&f(x)=x^{2}+2 x+1 \\
&g(x)=2 x+5
\end{aligned}
$$

Uma Kumari

Numerade Educator

Problem 3

Set up the definite integral that gives the area of the region.
$$
\begin{aligned}
&f(x)=x^{2}-4 x+3 \\
&g(x)=-x^{2}+2 x+3
\end{aligned}
$$

Uma Kumari

Numerade Educator

Problem 4

Set up the definite integral that gives the area of the region.
$$
\begin{aligned}
&f(x)=x^{2} \\
&g(x)=x^{3}
\end{aligned}
$$

Uma Kumari

Numerade Educator

Problem 5

Set up the definite integral that gives the area of the region.
$$
\begin{aligned}
&f(x)=3\left(x^{3}-x\right) \\
&g(x)=0
\end{aligned}
$$

Uma Kumari

Numerade Educator

Problem 6

Set up the definite integral that gives the area of the region.
$$
\begin{aligned}
&f(x)=(x-1)^{3} \\
&g(x)=x-1
\end{aligned}
$$

Uma Kumari

Numerade Educator

Problem 7

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral.
$$
\int_{0}^{4}\left[(x+1)-\frac{x}{2}\right] d x
$$

Gregory Higby

Numerade Educator

Problem 8

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral.
$$
\int_{-1}^{1}\left[\left(1-x^{2}\right)-\left(x^{2}-1\right)\right] d x
$$

Lucas Finney

Numerade Educator

Problem 9

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral.
$$
\int_{0}^{6}\left[4\left(2^{-x / 3}\right)-\frac{x}{6}\right] d x
$$

Uma Kumari

Numerade Educator

Problem 10

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral.
$$
\int_{2}^{3}\left[\left(\frac{x^{3}}{3}-x\right)-\frac{x}{3}\right] d x
$$

Sahil Patel

Numerade Educator

Problem 11

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral.
$$
\int_{-\pi / 3}^{\pi / 3}(2-\sec x) d x
$$

Uma Kumari

Numerade Educator

Problem 12

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral.
$$
\int_{-\pi / 4}^{\pi / 4}\left(\sec ^{2} x-\cos x\right) d x
$$

Gregory Higby

Numerade Educator

Problem 13

Find the area of the region by integrating (a) with respect to $x$ and (b) with respect to $y$.
$$
\begin{aligned}
&x=4-y^{2} \\
&x=y-2
\end{aligned}
$$

Uma Kumari

Numerade Educator

Problem 14

Find the area of the region by integrating (a) with respect to $x$ and (b) with respect to $y$.
$$
\begin{aligned}
&y=x^{2} \\
&y=6-x
\end{aligned}
$$

Uma Kumari

Numerade Educator

Problem 15

Determine which value best approximates the area of the region bounded by the graphs of $f$ and $g .$ (Make your selection on the basis of a sketch of the region and not by performing any calculations.)
$f(x)=x+1, \quad g(x)=(x-1)^{2}$
(a) $-2$
(b) 2
(c) 10
(d) 4
(e) $\underline{8}$

Gregory Higby

Numerade Educator

Problem 16

Determine which value best approximates the area of the region bounded by the graphs of $f$ and $g .$ (Make your selection on the basis of a sketch of the region and not by performing any calculations.)
$f(x)=2-\frac{1}{2} x, \quad g(x)=2-\sqrt{x}$
(a) 1
(b) 6
(c) $-3$
(d) 3
(e) 4

Gregory Higby

Numerade Educator

Problem 17

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
y=\frac{1}{2} x^{3}+2, y=x+1, x=0, x=2
$$

Uma Kumari

Numerade Educator

Problem 18

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
y=-\frac{3}{8} x(x-8), y=10-\frac{1}{2} x, x=2, x=8
$$

Uma Kumari

Numerade Educator

Problem 19

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
f(x)=x^{2}-4 x, g(x)=0
$$

Uma Kumari

Numerade Educator

Problem 20

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
f(x)=-x^{2}+4 x+1, g(x)=x+1
$$

Uma Kumari

Numerade Educator

Problem 21

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
f(x)=x^{2}+2 x+1, \quad g(x)=3 x+3
$$

Uma Kumari

Numerade Educator

Problem 22

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
f(x)=-x^{2}+4 x+2, g(x)=x+2
$$

Uma Kumari

Numerade Educator

Problem 23

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
y=x, \quad y=2-x, \quad y=0
$$

Uma Kumari

Numerade Educator

Problem 24

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
y=\frac{1}{x^{2}}, \quad y=0, x=1, x=5
$$

Uma Kumari

Numerade Educator

Problem 25

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
f(x)=\sqrt{3 x}+1, g(x)=x+1
$$

Uma Kumari

Numerade Educator

Problem 26

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
f(x)=\sqrt[3]{x-1}, g(x)=x-1
$$

Uma Kumari

Numerade Educator

Problem 27

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
f(y)=y^{2}, g(y)=y+2
$$

Uma Kumari

Numerade Educator

Problem 28

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
f(y)=y(2-y), \quad g(y)=-y
$$

Uma Kumari

Numerade Educator

Problem 29

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
f(y)=y^{2}+1, g(y)=0, \quad y=-1, \quad y=2
$$

Uma Kumari

Numerade Educator

Problem 30

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
f(y)=\frac{y}{\sqrt{16-y^{2}}}, g(y)=0, \quad y=3
$$

Uma Kumari

Numerade Educator

Problem 31

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
f(x)=\frac{10}{x}, x=0, y=2, y=10
$$

Uma Kumari

Numerade Educator

Problem 32

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.
$$
g(x)=\frac{4}{2-x}, \quad y=4, x=0
$$

Uma Kumari

Numerade Educator

Problem 33

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$$
f(x)=x\left(x^{2}-3 x+3\right), \quad g(x)=x^{2}
$$

Uma Kumari

Numerade Educator

Problem 34

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$$
f(x)=x^{3}-2 x+1, g(x)=-2 x, x=1
$$

Uma Kumari

Numerade Educator

Problem 35

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$$
y=x^{2}-4 x+3, \quad y=3+4 x-x^{2}
$$

Uma Kumari

Numerade Educator

Problem 36

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$$
y=x^{4}-2 x^{2}, \quad y=2 x^{2}
$$

Uma Kumari

Numerade Educator

Problem 37

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$$
f(x)=x^{4}-4 x^{2}, g(x)=x^{2}-4
$$

Uma Kumari

Numerade Educator

Problem 38

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$$
f(x)=x^{4}-4 x^{2}, g(x)=x^{3}-4 x
$$

Uma Kumari

Numerade Educator

Problem 39

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$$
f(x)=1 /\left(1+x^{2}\right), \quad g(x)=\frac{1}{2} x^{2}
$$

Uma Kumari

Numerade Educator

Problem 40

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$$
f(x)=6 x /\left(x^{2}+1\right), y=0, \quad 0 \leq x \leq 3
$$

Uma Kumari

Numerade Educator

Problem 41

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$$
y=\sqrt{1+x^{3}}, y=\frac{1}{2} x+2, x \equiv 0
$$

Uma Kumari

Numerade Educator

Problem 42

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$$
y=x \sqrt{\frac{4-x}{4+x}}, y=0, x=4
$$

Uma Kumari

Numerade Educator

Problem 43

Sketch the region bounded by the graphs of the functions, and find the area of the region.
$f(x)=2 \sin x, g(x)=\tan x,-\frac{\pi}{3} \leq x \leq \frac{\pi}{3}$

Uma Kumari

Numerade Educator

Problem 44

Sketch the region bounded by the graphs of the functions, and find the area of the region.
$f(x)=\sin x, g(x)=\cos 2 x,-\frac{\pi}{2} \leq x \leq \frac{\pi}{6}$

Uma Kumari

Numerade Educator

Problem 45

Sketch the region bounded by the graphs of the functions, and find the area of the region.
$f(x)=\cos x, \mathrm{~g}(x)=2-\cos x, 0 \leq x \leq 2 \pi$

Uma Kumari

Numerade Educator

Problem 46

Sketch the region bounded by the graphs of the functions, and find the area of the region.
$f(x)=\sec \frac{\pi x}{4} \tan \frac{\pi x}{4}, g(x)=(\sqrt{2}-4) x+4, \quad x=0$

Uma Kumari

Numerade Educator

Problem 47

Sketch the region bounded by the graphs of the functions, and find the area of the region.
$f(x)=x e^{-x^{2}}, \quad y=0, \quad 0 \leq x \leq 1$

Uma Kumari

Numerade Educator

Problem 48

Sketch the region bounded by the graphs of the functions, and find the area of the region.
$f(x)=3^{x}, \quad g(x)=2 x+1$

Uma Kumari

Numerade Educator

Problem 49

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$f(x)=2 \sin x+\sin 2 x, \quad y=0, \quad 0 \leq x \leq \pi$

Uma Kumari

Numerade Educator

Problem 50

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$f(x)=2 \sin x+\cos 2 x, \quad y=0, \quad 0<x \leq \pi$

Uma Kumari

Numerade Educator

Problem 51

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$f(x)=\frac{1}{x^{2}} e^{1 / x}, \quad y=0, \quad 1 \leq x \leq 3$

Uma Kumari

Numerade Educator

Problem 52

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$g(x)=\frac{4 \ln x}{x}, y=0, x=5$

Uma Kumari

Numerade Educator

Problem 53

In Exercises $\mathbf{5 3 - 5 6}$, of the equations, (b) explain why the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places.
$$
y=\sqrt{\frac{x^{3}}{4-x}}, y=0, \quad x=3
$$

Uma Kumari

Numerade Educator

Problem 54

In Exercises $\mathbf{5 3 - 5 6}$, of the equations, (b) explain why the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places.
$$
y=\sqrt{x} e^{x}, \quad y=0, x=0, \quad x=1
$$

Uma Kumari

Numerade Educator

Problem 55

In Exercises $\mathbf{5 3 - 5 6}$, of the equations, (b) explain why the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places.
$$
y=x^{2}, \quad y=4 \cos x
$$

Uma Kumari

Numerade Educator

Problem 56

In Exercises $\mathbf{5 3 - 5 6}$, of the equations, (b) explain why the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places.
$$
y=x^{2}, \quad y=\sqrt{3+x}
$$

Uma Kumari

Numerade Educator

Problem 57

Find the accumulation function $F$. Then evaluate $F$ at each value of the independent variable and graphically show the area given by each value of $F$.
$F(x)=\int_{0}^{x}\left(\frac{1}{2} t+1\right) d t \quad$ (a) $F(0) \quad$ (b) $F(2) \quad$ (c) $F(6)$

Uma Kumari

Numerade Educator

Problem 58

Find the accumulation function $F$. Then evaluate $F$ at each value of the independent variable and graphically show the area given by each value of $F$.
$F(x)=\int_{0}^{x}\left(\frac{1}{2} t^{2}+2\right) d t \quad$ (a) $F(0) \quad$ (b) $F(4) \quad$ (c) $F(6)$

Uma Kumari

Numerade Educator

Problem 59

Find the accumulation function $F$. Then evaluate $F$ at each value of the independent variable and graphically show the area given by each value of $F$.
$F(\alpha)=\int_{-1}^{\alpha} \cos \frac{\pi \theta}{2} d \theta \quad$ (a) $F(-1) \quad$ (b) $F(0)$
(c) $F\left(\frac{1}{2}\right)$

Uma Kumari

Numerade Educator

Problem 60

Find the accumulation function $F$. Then evaluate $F$ at each value of the independent variable and graphically show the area given by each value of $F$.
$F(y)=\int_{-1}^{y} 4 e^{x / 2} d x \quad$ (a) $F(-1) \quad$ (b) $F(0)$ (c) $F(4)$

Uma Kumari

Numerade Educator

Problem 61

Use integration to find the area of the figure having the given vertices.
$$
(2,-3),(4,6),(6,1)
$$

Uma Kumari

Numerade Educator

Problem 62

Use integration to find the area of the figure having the given vertices.
$$
(0,0),(a, 0),(b, c)
$$

Uma Kumari

Numerade Educator

Problem 63

Use integration to find the area of the figure having the given vertices.
$$
(0,2),(4,2),(0,-2),(-4,-2)
$$

Uma Kumari

Numerade Educator

Problem 64

Use integration to find the area of the figure having the given vertices.
$$
(0,0),(1,2),(3,-2),(1,-3)
$$

Uma Kumari

Numerade Educator

Problem 65

Estimate the surface area of the golf green using (a) the Trapezoidal Rule and (b) Simpson's Rule.

Uma Kumari

Numerade Educator

Problem 66

Estimate the surface area of the oil spill using (a) the Trapezoidal Rule and (b) Simpson's Rule.

Uma Kumari

Numerade Educator

Problem 67

Set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point.
$$
f(x)=x^{3}, \quad(1,1)
$$

Gregory Higby

Numerade Educator

Problem 68

Set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point.
$$
y=x^{3}-2 x, \quad(-1,1)
$$

Uma Kumari

Numerade Educator

Problem 69

Set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point.
$$
f(x)=\frac{1}{x^{2}+1}, \quad\left(1, \frac{1}{2}\right)
$$

Uma Kumari

Numerade Educator

Problem 70

Set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point.
$$
y=\frac{2}{1+4 x^{2}}, \quad\left(\frac{1}{2}, 1\right)
$$

Uma Kumari

Numerade Educator

Problem 71

The graphs of $y=x^{4}-2 x^{2}+1$ and $y=1-x^{2}$ intersect at three points. However, the area between the curves can be found by a single integral. Explain why this is so, and write an integral for this area.

Uma Kumari

Numerade Educator

Problem 72

The area of the region bounded by the graphs of $y=x^{3}$ and $y=x$ cannot be found by the single integral $\int_{-1}^{1}\left(x^{3}-x\right) d x .$ Explain why this is so. Use symmetry to write a single integral that does represent the area.

Sahil Patel

Numerade Educator

Problem 73

A college graduate has two job offers. The starting salary for each is $$\$ 32,000$$, and after 8 years of service each will pay $$\$ 54,000$$. The salary increase for each offer is shown in the figure. From a strictly monetary viewpoint, which is the better offer? Explain.

Gregory Higby

Numerade Educator

Problem 74

A state legislature is debating two proposals for eliminating the annual budget deficits by the year $2010 .$ The rate of decrease of the deficits for each proposal is shown in the figure. From the viewpoint of minimizing the cumulative state deficit, which is the better proposal? Explain.

Lucas Finney

Numerade Educator

Problem 75

Find $b$ such that the line $y=b$ divides the region bounded by the graphs of the two equations into two regions of equal area.
$$
y=9-x^{2}, \quad y=0
$$

Gregory Higby

Numerade Educator

Problem 76

Find $b$ such that the line $y=b$ divides the region bounded by the graphs of the two equations into two regions of equal area.
$$
y=9-|x|, \quad y=0
$$

Uma Kumari

Numerade Educator

Problem 77

Find $a$ such that the line $x=a$ divides the region bounded by the graphs of the equations into two regions of equal area.
$$
y=x, \quad y=4, \quad x=0
$$

Gregory Higby

Numerade Educator

Problem 78

Find $a$ such that the line $x=a$ divides the region bounded by the graphs of the equations into two regions of equal area.
$$
y^{2}=4-x, \quad x=0
$$

Uma Kumari

Numerade Educator

Problem 79

Evaluate the limit and sketch the graph of the region whose area is represented by the limit.
$\lim _{\|\Delta\| 0} \sum_{i=1}^{n}\left(x_{i}-x_{i}^{2}\right) \Delta x$, where $x_{i}=i / n$ and $\Delta x=1 / n$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 80

Evaluate the limit and sketch the graph of the region whose area is represented by the limit.
$\lim _{\| \Delta \rightarrow 0} \sum_{i=1}^{n}\left(4-x_{i}^{2}\right) \Delta x$, where $x_{i}=-2+(4 i / n)$ and $\Delta x=4 / n$

Sahil Patel

Numerade Educator

Problem 81

Two models $R_{1}$ and $R_{2}$ are given for revenue (in billions of dollars per year) for a large corporation. The model $R_{1}$ gives projected annual revenues from 2000 to 2005, with $t=0$ corresponding to 2000, and $R_{2}$ gives projected revenues if there is a decrease in the rate of growth of corporate sales over the period. Approximate the total reduction in revenue if corporate sales are actually closer to the model $\boldsymbol{R}_{\mathbf{2}}$
$$
\begin{aligned}
&R_{1}=7.21+0.58 t \\
&R_{2}=7.21+0.45 t
\end{aligned}
$$

Uma Kumari

Numerade Educator

Problem 82

Two models $R_{1}$ and $R_{2}$ are given for revenue (in billions of dollars per year) for a large corporation. The model $R_{1}$ gives projected annual revenues from 2000 to 2005, with $t=0$ corresponding to 2000, and $R_{2}$ gives projected revenues if there is a decrease in the rate of growth of corporate sales over the period. Approximate the total reduction in revenue if corporate sales are actually closer to the model $\boldsymbol{R}_{\mathbf{2}}$
$$
\begin{aligned}
&R_{1}=7.21+0.26 t+0.02 t^{2} \\
&R_{2}=7.21+0.1 t+0.01 t^{2}
\end{aligned}
$$

Uma Kumari

Numerade Educator

Problem 83

The table shows the total receipts $R$ and total expenditures $E$ for the Old-Age and Survivors Insurance Trust Fund (Social Security Trust Fund) in billions of dollars. The time $t$ is given in years, with $t=1$ corresponding to 1991 .
$$
\begin{array}{|l|c|c|c|c|c|c|}
\hline \boldsymbol{t} & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline \boldsymbol{R} & 299.3 & 311.2 & 323.3 & 328.3 & 342.8 & 363.7 \\
\hline \boldsymbol{E} & 245.6 & 259.9 & 273.1 & 284.1 & 297.8 & 308.2 \\
\hline
\end{array}
$$
$$
\begin{array}{|l|c|c|c|c|c|}
\hline \boldsymbol{t} & 7 & 8 & 9 & 10 & 11 \\
\hline \boldsymbol{R} & 397.2 & 424.8 & 457.0 & 490.5 & 518.1 \\
\hline \boldsymbol{E} & 322.1 & 332.3 & 339.9 & 358.3 & 377.5 \\
\hline
\end{array}
$$
(a) Use a graphing utility to fit an exponential model to the data for receipts. Plot the data and graph the model.
(b) Use a graphing utility to fit an exponential model to the data for expenditures. Plot the data and graph the model.
(c) If the models are assumed to be true for the years 2002 through 2007, use integration to approximate the surplus revenue generated during those years.
(d) Will the models found in parts (a) and
(b) intersect? Explain. Based on your answer and news reports about the fund, will these models be accurate for long-term analysis?

Uma Kumari

Numerade Educator

Problem 84

Economists use Lorenz curves to illustrate the distribution of income in a country. A Lorenz curve, $y=f(x)$, represents the actual income distribution in the country. In this model, $x$ represents percents of families in the country and $y$ represents percents of total income. The model $y=x$ represents a country in which each family has the same income. The area between these two models, where $0 \leq x \leq 100$, indicates a country's "income inequality." The table lists percents of income $y$ for selected percents of families $x$ in a country.
$$
\begin{aligned}
&\begin{array}{|c|c|c|c|c|c|}
\hline x & 10 & 20 & 30 & 40 & 50 \\
\hline y & 3.35 & 6.07 & 9.17 & 13.39 & 19.45 \\
\hline
\end{array}\\
&\begin{array}{|c|c|c|c|c|}
\hline x & 60 & 70 & 80 & 90 \\
\hline y & 28.03 & 39.77 & 55.28 & 75.12 \\
\hline
\end{array}
\end{aligned}
$$
(a) Use a graphing utility to find a quadratic model for the Lorenz curve.
(b) Plot the data and graph the model.
(c) Graph the model $y=x .$ How does this model compare with the model in part (a)?
(d) Use the integration capabilities of a graphing utility to approximate the "income inequality."

Uma Kumari

Numerade Educator

Problem 85

The chief financial officer of a company reports that profits for the past fiscal year were $$\$ 893,000 .$$ The officer predicts that profits for the next 5 years will grow at a continuous annual rate somewhere between $3 \frac{1}{2} \%$ and $5 \% .$ Estimate the cumulative difference in total profit over the 5 years based on the predicted range of growth rates.

Uma Kumari

Numerade Educator

Problem 86

The shaded region in the figure consists of all points whose distances from the center of the square are less than their distances from the edges of the square. Find the area of the region.

Uma Kumari

Numerade Educator

Problem 87

The surface of a machine part is the region between the graphs of $y_{1}=|x|$ and $y_{2}=0.08 x^{2}+k$ (see figure).
(a) Find $k$ if the parabola is tangent to the graph of $y_{1}$.
(b) Find the area of the surface of the machine part.

Uma Kumari

Numerade Educator

Problem 88

Concrete sections for a new building have the dimensions (in meters) and shape shown in the figure.
(a) Find the area of the face of the section superimposed on the rectangular coordinate system.
(b) Find the volume of concrete in one of the sections by multiplying the area in part (a) by 2 meters.
(c) One cubic meter of concrete weighs 5000 pounds. Find the weight of the section.

Uma Kumari

Numerade Educator

Problem 89

To decrease the weight and to aid in the hardening process, the concrete sections in Exercise 88 often are not solid. Rework Exercise 88 to allow for cylindrical openings such as those shown in the figure.

Uma Kumari

Numerade Educator

Problem 90

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If the area of the region bounded by the graphs of $f$ and $g$ is 1, then the area of the region bounded by the graphs of $h(x)=f(x)+C$ and $k(x)=g(x)+C$ is also $1 .$

Gregory Higby

Numerade Educator

Problem 91

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
$$
\text { If } \int_{a}^{b}[f(x)-g(x)] d x=A, \text { then } \int_{a}^{b}[g(x)-f(x)] d x=-A
$$

Uma Kumari

Numerade Educator

Problem 92

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If the graphs of $f$ and $g$ intersect midway between $x=a$ and $x=b$, then
$\int_{a}^{b}[f(x)-g(x)] d x=0$

Gregory Higby

Numerade Educator

Problem 93

Find the area between the graph of $y=\sin x$ and the line segments joining the points $(0,0)$ and $\left(\frac{7 \pi}{6},-\frac{1}{2}\right)$, as shown in the figure.

Uma Kumari

Numerade Educator

Problem 94

Let $a>0$ and $b>0$. Show that the area of the ellipse
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
is $\pi a b$ (see figure).

Uma Kumari

Numerade Educator

Problem 95

The horizontal line $y=c$ intersects the curve $y=2 x-3 x^{3}$ in the first quadrant as shown in the figure. Find $c$ so that the areas of the two shaded regions are equal.

Gregory Higby

Numerade Educator