Calculus, Early Transcendentals

Michael Sullivan, Kathleen Miranda

Chapter 6

Applications of the Integral - all with Video Answers

Educators

Section 1

Area Between Graphs

Problem 1

Express the area between the graphs of $y=x^{2}$ and $y=\sqrt{x}$ as an integral using a partition of the $x$ -axis. Do not find the integral.

Adrian Co

Adrian Co

Numerade Educator

Problem 2

Express the area between the graph of $x=y^{2}$ and the line $x=1$ as an integral using a partition of the $y$ -axis. Do not find the integral.

Adrian Co

Numerade Educator

Problem 3

In Problems 3-12, find the area of the region enclosed by the graphs of the given equations by partitioning the $x$ -axis.
$$
y=x, y=2 x, x=1
$$

Adrian Co

Numerade Educator

Problem 4

Find the area of the region enclosed by the graphs of the given equations by partitioning the $x$ -axis.
$$
y=x, y=3 x, x=3
$$

Adrian Co

Numerade Educator

Problem 5

Find the area of the region enclosed by the graphs of the given equations by partitioning the $x$ -axis.
$$
y=x^{2}, y=x
$$

Adrian Co

Numerade Educator

Problem 6

Find the area of the region enclosed by the graphs of the given equations by partitioning the $x$ -axis.
$$
y=x^{2}, y=4 x
$$

Adrian Co

Numerade Educator

Problem 7

Find the area of the region enclosed by the graphs of the given equations by partitioning the $x$ -axis.
$$
y=e^{x}, y=e^{-x}, x=\ln 2
$$

Adrian Co

Numerade Educator

Problem 8

Find the area of the region enclosed by the graphs of the given equations by partitioning the $x$ -axis.
$$
y=e^{x}, y=-x+1, x=1
$$

Adrian Co

Numerade Educator

Problem 9

Find the area of the region enclosed by the graphs of the given equations by partitioning the $x$ -axis.
$$
y=x^{2}, y=x^{4}
$$

Adrian Co

Numerade Educator

Problem 10

Find the area of the region enclosed by the graphs of the given equations by partitioning the $x$ -axis.
$$
y=x, y=x^{3}
$$

Adrian Co

Numerade Educator

Problem 11

Find the area of the region enclosed by the graphs of the given equations by partitioning the $x$ -axis.
$$
y=\cos x, y=\frac{1}{2}, 0 \leq x \leq \frac{\pi}{3}
$$

Adrian Co

Numerade Educator

Problem 12

Find the area of the region enclosed by the graphs of the given equations by partitioning the $x$ -axis.
$$
y=\sin x, y=\frac{1}{2}, \frac{\pi}{6} \leq x \leq \frac{5 \pi}{6}
$$

Adrian Co

Numerade Educator

Problem 13

In Problems $13-20$, find the area of the region enclosed by the graphs of the given equations by partitioning the $y$ -axis.
$$
x=y^{2}, x=2-y
$$

Adrian Co

Numerade Educator

Problem 14

Find the area of the region enclosed by the graphs of the given equations by partitioning the $y$ -axis.
$$
x=y^{2}, x=y+2
$$

Adrian Co

Numerade Educator

Problem 15

Find the area of the region enclosed by the graphs of the given equations by partitioning the $y$ -axis.
$$
x=9-y^{2}, x=5
$$

Adrian Co

Numerade Educator

Problem 16

Find the area of the region enclosed by the graphs of the given equations by partitioning the $y$ -axis.
$$
x=16-y^{2}, x=7
$$

Adrian Co

Numerade Educator

Problem 17

Find the area of the region enclosed by the graphs of the given equations by partitioning the $y$ -axis.
$$
x=y^{2}+4, y=x-6
$$

Adrian Co

Numerade Educator

Problem 18

Find the area of the region enclosed by the graphs of the given equations by partitioning the $y$ -axis.
$$
x=y^{2}+6, y=8-x
$$

Adrian Co

Numerade Educator

Problem 19

Find the area of the region enclosed by the graphs of the given equations by partitioning the $y$ -axis.
$$
y=\ln x, x=1, y=2
$$

Adrian Co

Numerade Educator

Problem 20

Find the area of the region enclosed by the graphs of the given equations by partitioning the $y$ -axis.
$$
y=\ln x, x=e, y=0
$$

Adrian Co

Numerade Educator

Problem 21

In Problems 21-24, find the area of the shaded region in the graph.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 22

In Problems $21-24$, find the area of the shaded region in the graph.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 23

In Problems $21-24$, find the area of the shaded region in the graph.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 24

In Problems $21-24$, find the area of the shaded region in the graph.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 25

In Problems 25-32, find the area A of the region enclosed by the graphs of the given equations:
(a) by partitioning the $x$ -axis.
(b) by partitioning the $y$ -axis.
$$
y=\sqrt{x}, y=x^{3}
$$

Supratim Pal

Numerade Educator

Problem 26

Find the area A of the region enclosed by the graphs of the given equations:
(a) by partitioning the $x$ -axis.
(b) by partitioning the $y$ -axis.
$$
y=\sqrt{x}, y=x^{2}
$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 27

Find the area A of the region enclosed by the graphs of the given equations:
(a) by partitioning the $x$ -axis.
(b) by partitioning the $y$ -axis.
$$
y=x^{2}+1, y=x+1
$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 28

Find the area A of the region enclosed by the graphs of the given equations:
(a) by partitioning the $x$ -axis.
(b) by partitioning the $y$ -axis.
$$
y=x^{2}+1, y=4 x+1
$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 29

Find the area A of the region enclosed by the graphs of the given equations:
(a) by partitioning the $x$ -axis.
(b) by partitioning the $y$ -axis.
$$
y=\sqrt{9-x}, y=\sqrt{9-3 x}, x \text { -axis }
$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 30

Find the area A of the region enclosed by the graphs of the given equations:
(a) by partitioning the $x$ -axis.
(b) by partitioning the $y$ -axis.
$$
y=\sqrt{16-2 x}, y=\sqrt{16-4 x}, x \text { -axis }
$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 31

Find the area A of the region enclosed by the graphs of the given equations:
(a) by partitioning the $x$ -axis.
(b) by partitioning the $y$ -axis.
$$
y=\sqrt{2 x-6}, y=\sqrt{x-2}, x \text { -axis }
$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 32

Find the area A of the region enclosed by the graphs of the given equations:
(a) by partitioning the $x$ -axis.
(b) by partitioning the $y$ -axis.
$$
y=\sqrt{2 x-5}, y=\sqrt{4 x-17}, x \text { -axis }
$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 33

In Problems 33-46, find the area of the region enclosed by the graphs of the given equations.
$$
y=4-x^{2}, y=x^{2}
$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 34

Find the area of the region enclosed by the graphs of the given equations.
$$
y=9-x^{2}, y=x^{2}
$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 35

Find the area of the region enclosed by the graphs of the given equations.
$$
x=y^{2}-4, x=4-y^{2}
$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 36

Find the area of the region enclosed by the graphs of the given equations.
$$
x=y^{2}, x=16-y^{2}
$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 37

Find the area of the region enclosed by the graphs of the given equations.
$y=\ln x^{2},$ the $x$ -axis, and the line $x=e$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 38

Find the area of the region enclosed by the graphs of the given equations.
$y=\ln x, y=1-x,$ and the line $y=1$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 39

Find the area of the region enclosed by the graphs of the given equations.
$y=\cos x, y=1-\frac{3}{\pi} x, x=\frac{\pi}{3}$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 40

Find the area of the region enclosed by the graphs of the given equations.
$y=\sin x, y=1,0 \leq x \leq \frac{\pi}{2}$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 41

Find the area of the region enclosed by the graphs of the given equations.
$y=e^{2 x}$ and the lines $x=1$ and $y=1$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 42

Find the area of the region enclosed by the graphs of the given equations.
$y=e^{x}, \quad y=e^{3 x}, x=2$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 43

Find the area of the region enclosed by the graphs of the given equations.
$y^{2}=4 x, 4 x-3 y-4=0$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 44

Find the area of the region enclosed by the graphs of the given equations.
$y^{2}=4 x+1, x=y+1$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 45

Find the area of the region enclosed by the graphs of the given equations.
$y=\sin x, y=\frac{2 x}{\pi}, x \geq 0$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 46

Find the area of the region enclosed by the graphs of the given equations.
$y=\cos x, x \geq 0, y=\frac{3 x}{\pi}$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 47

An Archimedean Result
Show that the area of the shaded region in the figure is two-thirds of the area of the parallelogram $A B C D$. (This illustrates a result due to Archimedes concerning sectors of parabolas.)

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 48

Equal Areas Find $h$ so that the area of the region enclosed by the graphs of $y=x$ $y=8 x,$ and $y=\frac{1}{x^{2}}$ is equal to that of an isosceles triangle of base 1 and height $h .$ See the figure.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 49

Cost of Health Care The cost of health care varies from one country to another. Between 2000 and $2010,$ the average cost of health insurance for a family of four in the United States was modeled by
$$
A(x)=8020.6596\left(1.0855^{x}\right) \quad 0 \leq x \leq 10
$$
where $x=0$ corresponds to the year 2000 and $A(x)$ is measured in U.S. dollars. During the same years, the average cost of health care in Canada was given by
$$
C(x)=4944.6424\left(1.0711^{x}\right) \quad 0 \leq x \leq 10
$$
where $x=0$ corresponds to the year 2000 and $C(x)$ is measured in U.S. dollars.
(a) Find the area between the graphs from $x=0$ to $x=10$. Round the answer to the nearest dollar.
(b) Interpret the answer to (a).

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 50

Area Find the area in the first quadrant enclosed by the graphs of $y=\sin (2 x)$ and $y=\cos (2 x), 0 \leq x \leq \frac{\pi}{8}$.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 51

Area Find the area of the region enclosed by the graphs of $y=\sin ^{-1} x, y=x, 0 \leq x \leq \frac{1}{2} .$ Which axis did you choose to partition? Explain your choice.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 52

Area
(a) Graph $y=x^{2}$ and $y=\sin x$
(b) Find the points of intersection of the graphs.
(c) Find the area enclosed by the graphs of $y=x^{2}$ and $y=\sin x$ using a partition of the $x$ -axis.
(d) Find the area enclosed by the graphs of $y=x^{2}$ and $y=\sin x$ using a partition of the $y$ -axis.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 53

(a) Graph $y=\sin ^{-1} x, x+y=1,$ and $y=0$
(b) Find the points of intersection of the graphs.
(c) Find the area enclosed by the graphs.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 54

(a) Graph $y=\cos ^{-1} x, y=x^{3}+1,$ and $y=0$.
(b) Find the points of intersection of the graphs.
(c) Find the area enclosed by the graphs.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 55

(a) $\operatorname{Graph} 2 x+y=3, \quad y=\frac{1}{1+x^{2}},$ and $y=1$.
(b) Find the points of intersection of the graphs.
(c) Find the area enclosed by the graphs.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 56

(a) Graph $y=\frac{5}{1+x^{2}}, x=\sqrt{y+2},$ and $x=0$.
(b) Find the points of intersection of the graphs.
(c) Find the area enclosed by the graphs.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 57

(a) Graph $y=\sin ^{-1} x, y=1-x^{2},$ and $y=0$.
(b) Find the points of intersection of the graphs.
(c) Find the area enclosed by the graphs.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 58

Find the area enclosed by the graph of $y^{2}=x^{2}-x^{4}$.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 59

(a) Express the area $A$ of the region in the first quadrant enclosed by the $y$ -axis and the graphs of $y=\tan x$ and $y=k$ for $k>0$ as a function of $k$.
(b) What is the value of $A$ when $k=1 ?$
(c) If the line $y=k$ is moving upward at the rate of $\frac{1}{10}$ unit per second, at what rate is $A$ changing when $k=1 ?$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 60

The area $A$ of the shaded region in the figure is $A=\frac{t}{2} .$ Prove this as follows:
(a) Let $A^{*}$ be twice the area outlined in the figure, and $(x, y)$ be the coordinates of $P$. Explain why
$$
A^{*}=x \sqrt{x^{2}-1}-2 \int_{1}^{x} \sqrt{u^{2}-1} d u
$$
(b) Differentiate $A^{*}$ to show that $\frac{d A^{*}}{d x}=\frac{1}{\sqrt{x^{2}-1}}$
(c) Show that $A^{*}=\cosh ^{-1} x+C,$ and explain why $C=0$.
(d) Why does it follow that $A^{*}=t ?$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator