Calculus

Jon Rogawski, Jonathan David Rogawski

Chapter 6

APPLICATIONS OF THE INTEGRAL - all with Video Answers

Educators

Section 1

Area Between Two Curves

Problem 1

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

Check back soon!

Problem 3

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 Jiang

Numerade Educator

Problem 3

Find the area of the region enclosed by the graphs of $f(x)=x^2+2$ and $g(x)=2 x+5$ (Figure 10).
FIGURE CANT COPY

Linh Vu

Numerade Educator

Problem 4

Find the area of the region enclosed by the graphs of $f(x)= x^3-10 x$ and $g(x)=6 x$ (Figure 11).
FIGURE CANT COPY

Gregory Higby

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 Vu

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.
$[0, \pi]$

Linh Vu

Numerade Educator

Problem 7

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

Gregory Higby

Numerade Educator

Problem 8

In Exercises 7 and 8, let $f(x)=20+x-x^2$ and $g(x)=x^2-5 x$.
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.

LB

La B

Numerade Educator

Problem 9

GU Find the points of intersection of $y=x\left(x^2-1\right)$ and $y= 1-x^2$. Sketch the region enclosed by these curves over $[-1,1]$ and compute its area.

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 10

GU Find the points of intersection of $y=x(4-x)$ and $y= x^2(4-x)$. Sketch the region enclosed by these curves over $[0,4]$ and compute its area.

Check back soon!

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.

Willis James

Numerade Educator

Problem 12

Sketch the region bounded by

$$
y=\frac{x}{\sqrt{1-x^2}} \quad \text { and } \quad y=-\frac{x}{\sqrt{1-x^2}}
$$

for $0 \leq x \leq 0.8$ and find its area.

Amy Jiang

Numerade Educator

Problem 13

In Exercises 13-16, find the area of the shaded region in Figures 12-15.
FIGURE CANT COPY

Linh Vu

Numerade Educator

Problem 14

In Exercises 13-16, find the area of the shaded region in Figures 12-15.
FIGURE CANT COPY

Linh Vu

Numerade Educator

Problem 15

In Exercises 13-16, find the area of the shaded region in Figures 12-15.
FIGURE CANT COPY

Linh Vu

Numerade Educator

Problem 16

In Exercises 13-16, find the area of the shaded region in Figures 12-15.
FIGURE CANT COPY

Linh Vu

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 Vu

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).
$-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$
FIGURE CANT COPY

Linh Vu

Numerade Educator

Problem 19

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

Colin O'Haire

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 Jiang

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.
FIGURE CANT COPY

Gregory Higby

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.
FIGURE CANT COPY Region between $y=x^3-6 x$ and $y=8-3 x^2$.

Amy Jiang

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 Vu

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.
The semicubical parabola $y^2=x^3$ and the line $x=1$.

Amy Jiang

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.
Region between $y^2=x+5$ and $y^2=3-x$

Linh Vu

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.
Region between $y=x$ and $x+y=8$ over $[2,3]$

Amy Jiang

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.
$y=4-x^2, \quad y=x^2-4$

Gregory Higby

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.
$y=x^2-6, \quad y=6-x^3, \quad y$-axis

Gregory Higby

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.
$x+y=4, \quad x-y=0, \quad y+3 x=4$

Gregory Higby

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 Jiang

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.
$y=8-\sqrt{x}, \quad y=\sqrt{x}, \quad x=0$

Gregory Higby

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.
$y=\left|x^2-4\right|, \quad y=5$

Linh Vu

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 Vu

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.
$y=|x|, \quad y=x^2-6$

Gregory Higby

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.
$x=y^3-18 y, \quad y+2 x=0$

Gregory Higby

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 Jiang

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.
$x=2 y, \quad x+1=(y-1)^2$

Gregory Higby

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.
$x+y=1, \quad x^{1 / 2}+y^{1 / 2}=1$

Gregory Higby

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 Vu

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 Jiang

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 Vu

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.
$x=\sin y, \quad x=\frac{2}{\pi} y$

Linda Hand

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.
$y=\sin x, \quad y=x \sin \left(x^2\right), \quad 0 \leq x \leq 1$

Amy Jiang

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.
$y=\frac{\sin (\sqrt{x})}{\sqrt{x}}, \quad y=0, \quad \pi^2 \leq x \leq 9 \pi^2$

Amy Jiang

Numerade Educator

Problem 45

Plot

$$
y=\frac{x}{\sqrt{x^2+1}} \text { and } y=(x-1)^2
$$

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.

Bobby Barnes

University of North Texas

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 Jiang

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.
(a) Which of the following is represented by the area of the shaded region over $[0,10]$ ?
i. The distance between athletes 1 and 2 at time $t=10 \mathrm{~s}$.
ii. The difference in the distance traveled by the athletes over the time interval $[0,10]$.
(b) Does Figure 19 give us enough information to determine who is ahead at time $t=10 \mathrm{~s}$ ?
(c) If the athletes begin at the same time and place, who is ahead at $t=10 \mathrm{~s}$ ? At $t=25 \mathrm{~s}$ ?
FIGURE CANT COPY

Gregory Higby

Numerade Educator

Problem 48

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

Linda Hand

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 .

Linda Hand

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 Jiang

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 Vu

Numerade Educator

Problem 52

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

Amy Jiang

Numerade Educator

Problem 53

$2^{-} 95$ Find a numerical approximation to the area above $y=|x|$ and below $y=\cos x$.

Linh Vu

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 Jiang

Numerade Educator

Problem 55

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

$$
\begin{aligned}
& 6,9,10.25,10.75,10.75,10.25,9.75,9.5,10,11.25, \\
& 12.75,13.75,14.25,14.5,14.5,14,13.25,11.25,9
\end{aligned}
$$
Use the midpoint rule to estimate the area of the back.
FIGURE CANT COPY

Gregory Higby

Numerade Educator

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 Jiang

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 $r$ th 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 $[0,1]$.
(a) Show that $G=2 \int_0^1(r-F(r)) d r$.
(b) Calculate $G$ if

$$
F(r)= \begin{cases}\frac{1}{3} r & \text { for } 0 \leq r \leq \frac{1}{2} \\ \frac{5}{3} r-\frac{2}{3} & \text { for } \frac{1}{2} \leq r \leq 1\end{cases}
$$

(c) The Gini index is a measure of income distribution, with a lower value indicating a more equal distribution. Calculate $G$ if $F(r)=r$ (in this case, all households have the same income by Exercise 51(b) of Section 3.4).
(d) What is $G$ if all of the income goes to one household? Hint: In this extreme case, $F(r)=0$ for $0 \leq r<1$.

Amy Jiang

Numerade Educator

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 $r$ th fraction of households. The graph of $y=F(r)$ is called the Lorenz curve.

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.
$$
\begin{array}{|l|c|c|c|c|c|c|}
\hline r & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \\
\hline F(r) & 0 & 0.035 & 0.123 & 0.269 & 0.499 & 1 \\
\hline
\end{array}
$$
FIGURE CANT COPY

Amy Jiang

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.

Yuki Hotta

Numerade Educator

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.
FIGURE CANT COPY

Amy Jiang

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 Vu

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 Jiang

Numerade Educator