Calculus: Early Transcendentals, Metric Edition

James Stewart, Daniel K. Clegg, Saleem Watson

Chapter 6

Applications of Integration - all with Video Answers

Educators

+ 1 more educators

Section 1

Areas Between Curves

Problem 1

(a) Set up an integral for the area of the shaded region.
(b) Evaluate the integral to find the area.

Gregory Higby

Gregory Higby

Numerade Educator

Problem 2

(a) Set up an integral for the area of the shaded region.
(b) Evaluate the integral to find the area.

Gregory Higby

Numerade Educator

Problem 3

(a) Set up an integral for the area of the shaded region.
(b) Evaluate the integral to find the area.

Gregory Higby

Numerade Educator

Problem 4

(a) Set up an integral for the area of the shaded region.
(b) Evaluate the integral to find the area.

Gregory Higby

Numerade Educator

Problem 5

5-6 Find the area of the shaded region.

Gregory Higby

Numerade Educator

Problem 6

Find the area of the shaded region.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 7

7-10 Set up, but do not evaluate, an integral representing the area of the region enclosed by the given curves.
$$
y=2^{x}, \quad y=3^{x}, \quad x=1
$$

Gregory Higby

Numerade Educator

Problem 8

Set up, but do not evaluate, an integral representing the area of the region enclosed by the given curves.
$$
y=\ln x, \quad y=\ln \left(x^{2}\right), \quad x=2
$$

Gregory Higby

Numerade Educator

Problem 9

Set up, but do not evaluate, an integral representing the area of the region enclosed by the given curves.
$$
y=2-x, \quad y=2 x-x^{2}
$$

Gregory Higby

Numerade Educator

Problem 10

Set up, but do not evaluate, an integral representing the area of the region enclosed by the given curves.
$$
x=y^{4}, \quad x=2-y^{2}
$$

Gregory Higby

Numerade Educator

Problem 11

$11-18$ Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y .$ Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$$
y=x^{2}+2, \quad y=-x-1, \quad x=0, \quad x=1
$$

Khushbu Rani

Numerade Educator

Problem 12

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y .$ Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$$
y=1+x^{3}, \quad y=2-x, \quad x=-1, \quad x=0
$$

Gregory Higby

Numerade Educator

Problem 13

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y .$ Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$$
y=1 / x, \quad y=1 / x^{2}, \quad x=2
$$

SL

Sky Li

Numerade Educator

Problem 14

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y .$ Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$$
y=\cos x, \quad y=e^{x}, \quad x=\pi / 2
$$

Adalynn Griesser

Adalynn Griesser

Numerade Educator

Problem 15

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y .$ Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$$
y=(x-2)^{2}, \quad y=x
$$

SL

Sky Li

Numerade Educator

Problem 16

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y .$ Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$$
y=x^{2}-4 x, y=2 x
$$

Sam Low

Numerade Educator

Problem 17

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y .$ Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$$
x=1-y^{2}, \quad x=y^{2}-1
$$

SL

Sky Li

Numerade Educator

Problem 18

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y .$ Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$$
4 x+y^{2}=12, x=y
$$

SL

Sky Li

Numerade Educator

Problem 19

19-36 Sketch the region enclosed by the given curves and find its area.
$$
y=12-x^{2}, \quad y=x^{2}-6
$$

Gregory Higby

Numerade Educator

Problem 20

Sketch the region enclosed by the given curves and find its area.
$$
y=x^{2}, \quad y=4 x-x^{2}
$$

Fuzail Shakir

Numerade Educator

Problem 21

Sketch the region enclosed by the given curves and find its area.
$$
x=2 y^{2}, \quad x=4+y^{2}
$$

SL

Sky Li

Numerade Educator

Problem 22

Sketch the region enclosed by the given curves and find its area.
$$
y=\sqrt{x-1}, \quad x-y=1
$$

SL

Sky Li

Numerade Educator

Problem 23

Sketch the region enclosed by the given curves and find its area.
$$
y=\sqrt[3]{2 x}, \quad y=\frac{1}{2} x
$$

Gregory Higby

Numerade Educator

Problem 24

Sketch the region enclosed by the given curves and find its area.
$$
y=x^{3}, \quad y=x
$$

SL

Sky Li

Numerade Educator

Problem 25

Sketch the region enclosed by the given curves and find its area.
$$
y=\sqrt{x}, \quad y=\frac{1}{3} x, \quad 0 \leqslant x \leqslant 16
$$

Gregory Higby

Numerade Educator

Problem 26

Sketch the region enclosed by the given curves and find its area.
$$
y=\cos x, \quad y=2-\cos x, \quad 0 \leqslant x \leqslant 2 \pi
$$

Fuzail Shakir

Numerade Educator

Problem 27

Sketch the region enclosed by the given curves and find its area.
$$
y=\cos x, \quad y=\sin 2 x, \quad 0 \leqslant x=\pi / 2
$$

Adalynn Griesser

Adalynn Griesser

Numerade Educator

Problem 28

Sketch the region enclosed by the given curves and find its area.

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 29

Sketch the region enclosed by the given curves and find its area.
$$
y=\sec ^{2} x, \quad y=8 \cos x, \quad-\pi / 3 \leqslant x \leqslant \pi / 3
$$

Sam Low

Numerade Educator

Problem 30

Sketch the region enclosed by the given curves and find its area.
$$
y=x^{4}-3 x^{2}, \quad y=x^{2}
$$

Gregory Higby

Numerade Educator

Problem 31

Sketch the region enclosed by the given curves and find its area.
$$
y=x^{4}, \quad y=2-|x|
$$

Catherine Ross

Numerade Educator

Problem 32

Sketch the region enclosed by the given curves and find its area.
$$
y=x^{2}, \quad y=\frac{32}{x^{2}+4}
$$

Adalynn Griesser

Adalynn Griesser

Numerade Educator

Problem 33

Sketch the region enclosed by the given curves and find its area.
$$
y=\sin \frac{\pi x}{2}, \quad y=x^{3}
$$

Gregory Higby

Numerade Educator

Problem 34

Sketch the region enclosed by the given curves and find its area.
$$
y=4-2 \cosh x, \quad y=\frac{1}{2} \sinh x
$$

Andrija Isakov

Numerade Educator

Problem 35

Sketch the region enclosed by the given curves and find its area.
$$
y=1 / x, \quad y=x, \quad y=\frac{1}{4} x, \quad x>0
$$

Kenneth Kobos

Numerade Educator

Problem 36

Sketch the region enclosed by the given curves and find its area.
$$
y=\frac{1}{4} x^{2}, \quad y=2 x^{2}, \quad x+y=3, \quad x \geqslant 0
$$

Sam Low

Numerade Educator

Problem 37

The graphs of two functions are shown with the areas of the regions between the curves indicated.
(a) What is the total area between the curves for $0 \leqslant x \leqslant 5$ ?
(b) What is the value of $\int_{0}^{5}[f(x)-g(x)] d x$ ?

Mutahar Mehkri

Numerade Educator

Problem 38

$38-40$ Sketch the region enclosed by the given curves and find its
area.
$$
y=\frac{x}{\sqrt{1+x^{2}}}, \quad y=\frac{x}{\sqrt{9-x^{2}}}, \quad x \geqslant 0
$$

Adalynn Griesser

Adalynn Griesser

Numerade Educator

Problem 39

Sketch the region enclosed by the given curves and find its
area.
$$
y=\frac{x}{1+x^{2}}, \quad y=\frac{x^{2}}{1+x^{3}}
$$

Kenneth Kobos

Numerade Educator

Problem 40

Sketch the region enclosed by the given curves and find its
area.
$$
y=\frac{\ln x}{x}, \quad y=\frac{(\ln x)^{2}}{x}
$$

Kenneth Kobos

Numerade Educator

Problem 41

41-42 Use calculus to find the area of the triangle with the given vertices.
$$
(0,0), \quad(3,1), \quad(1,2)
$$

Adalynn Griesser

Adalynn Griesser

Numerade Educator

Problem 42

Use calculus to find the area of the triangle with the given vertices.
$$
(2,0), \quad(0,2), \quad(-1,1)
$$

Sam Low

Numerade Educator

Problem 43

43-44 Evaluate the integral and interpret it as the area of a region. Sketch the region.
$$
\int_{0}^{\pi / 2}|\sin x-\cos 2 x| d x
$$

Adalynn Griesser

Adalynn Griesser

Numerade Educator

Problem 44

Evaluate the integral and interpret it as the area of a region. Sketch the region.
$$
\int_{-1}^{1}\left|3^{x}-2^{x}\right| d x
$$

Andrija Isakov

Numerade Educator

Problem 45

45-48 Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.
$$
y=x \sin \left(x^{2}\right), \quad y=x^{4}, \quad x \geqslant 0
$$

Adalynn Griesser

Adalynn Griesser

Numerade Educator

Problem 46

Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.
$$
y=\frac{x}{\left(x^{2}+1\right)^{2}}, \quad y=x^{5}-x, \quad x \geqslant 0
$$

Sam Low

Numerade Educator

Problem 47

Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.
$$
y=3 x^{2}-2 x, \quad y=x^{3}-3 x+4
$$

Kenneth Kobos

Numerade Educator

Problem 48

Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.
$$
y=1.3^{x}, \quad y=2 \sqrt{x}
$$

Kenneth Kobos

Numerade Educator

Problem 49

49-52 Graph the region between the curves and compute the area correct to five decimal places.
$$
y=\frac{2}{1+x^{4}}, \quad y=x^{2}
$$

Adalynn Griesser

Adalynn Griesser

Numerade Educator

Problem 50

Graph the region between the curves and compute the area correct to five decimal places.
$$
y=e^{1-x^{2}}, \quad y=x^{4}
$$

Adalynn Griesser

Adalynn Griesser

Numerade Educator

Problem 51

Graph the region between the curves and compute the area correct to five decimal places.
$$
y=\tan ^{2} x, \quad y=\sqrt{x}
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 52

Graph the region between the curves and compute the area correct to five decimal places.
$$
y=\cos x, \quad y=x+2 \sin ^{4} x
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 53

Use a computer algebra system to find the exact area enclosed by the curves $y=x^{5}-6 x^{3}+4 x$ and $y=x$.

Kenneth Kobos

Numerade Educator

Problem 54

Sketch the region in the $x y$ -plane defined by the inequalities $x-2 y^{2} \geqslant 0,1-x-|y| \geqslant 0$ and find its area.

Sam Low

Numerade Educator

Problem 55

Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in kilometers per hour) during the first ten seconds of the race. Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.
$$
\begin{array}{|c|r|r||r|r|r|}
\hline t & v_{C} & v_{K} & t & v_{C} & v_{K} \\
\hline 0 & 0 & 0 & 6 & 110 & 128 \\
1 & 32 & 35 & 7 & 120 & 138 \\
2 & 51 & 59 & 8 & 130 & 150 \\
3 & 74 & 83 & 9 & 138 & 157 \\
4 & 86 & 98 & 10 & 144 & 163 \\
5 & 99 & 114 & & & \\
\hline
\end{array}
$$

James Kiss

Numerade Educator

Problem 56

The widths (in meters) of a kidney-shaped swimming pool were measured at 2 -meter intervals as indicated in the figure. Use the Midpoint Rule to estimate the area of the pool.

Bobby Barnes

University of North Texas

Problem 57

A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 20-centimeter intervals are $5.8,20.3,26.7,29.0,27.6$, $27.3,23.8,20.5,15.1,8.7$, and 2.8. Use the Midpoint Rule to estimate the area of the wing's cross-section.

Sam Low

Numerade Educator

Problem 58

If the birth rate of a population is $b(t)=2200 e^{a_{124 t}}$ people per year and the death rate is $d(t)=1460 e^{0.018 t}$ people per year, find the area between these curves for $0 \leqslant t \leqslant 10$. What does this area represent?

Gregory Higby

Numerade Educator

Problem 59

In Example 8 , we modeled a measles pathogenesis curve by a function $f .$ A patient infected with the measles virus who
has some immunity to the virus has a pathogenesis curve that can be modeled by, for instance, $g(t)=0.9 f(t)$.
(a) If the same threshold concentration of the virus is required for infectiousness to begin as in Example 8, on what day does this occur?
(b) Let $P_{3}$ be the point on the graph of $g$ where infectiousness begins. It has been shown that infectiousness ends at a point $P_{4}$ on the graph of $g$ where the line through $P_{3}, P_{4}$ has the same slope as the line through $P_{1}, P_{2}$ in Example $8(\mathrm{~b}) .$ On what day does infectiousness end?
(c) Compute the level of infectiousness for this patient.

Sam Low

Numerade Educator

Problem 60

The rates at which rain fell, in inches per hour, in two different locations $t$ hours after the start of a storm were modeled by $f(t)=0.73 t^{3}-2 t^{2}+t+0.6$ and $g(t)=0.17 t^{2}-0.5 t+1.1 .$ Compute the area between the graphs for $0 \leqslant t \leqslant 2$ and interpret your result in this context.

Andrija Isakov

Numerade Educator

Problem 61

The rates at which rain fell, in inches per hour, in two different locations $t$ hours after the start of a storm were modeled by $f(t)=0.73 t^{3}-2 t^{2}+t+0.6$ and $g(t)=0.17 t^{2}-0.5 t+1.1$. Compute the area between the graphs for $0 \leqslant t \leqslant 2$ and interpret your result in this context.

James Kiss

Numerade Educator

Problem 62

The figure shows graphs of the marginal revenue function $R^{\prime}$ and the marginal cost function $C^{\prime}$ for a manufacturer. [Recall from Section $4.7$ that $R(x)$ and $C(x)$ represent the revenue and cost when $x$ units are manufactured. Assume that $R$ and $C$ are measured in thousands of dollars.] What is the meaning of the area of the shaded region? Use the Midpoint Rule to estimate the value of this quantity.

James Kiss

Numerade Educator

Problem 63

The curve with equation $y^{2}=x^{2}(x+3)$ is called Tschirnhausen's cubic. If you graph this curve you will see that part of the curve forms a loop. Find the area enclosed by the loop.

Madi Sousa

Numerade Educator

Problem 64

Find the area of the region bounded by the parabola $y=x^{2}$, the tangent line to this parabola at $(1,1)$, and the $x$ -axis.

Fuzail Shakir

Numerade Educator

Problem 65

Find the number $b$ such that the line $y=b$ divides the region bounded by the curves $y=x^{2}$ and $y=4$ into two regions with equal area.

Catherine Ross

Numerade Educator

Problem 66

(a) Find the number $a$ such that the line $x=a$ bisects the area under the curve $y=1 / x^{2}, 1 \leqslant x \leqslant 4$.
(b) Find the number $b$ such that the line $y=b$ bisects the area in part (a).

Jocelyn Shackelford

Jocelyn Shackelford

Numerade Educator

Problem 67

Find the values of $c$ such that the area of the region bounded by the parabolas $y=x^{2}-c^{2}$ and $y=c^{2}-x^{2}$ is 576 .

Catherine Ross

Numerade Educator

Problem 68

Suppose that $0<c<\pi / 2$. For what value of $c$ is the area of the region enclosed by the curves $y=\cos x, y=\cos (x-c)$ and $x=0$ equal to the area of the region enclosed by the curves $y=\cos (x-c), x=\pi$, and $y=0$ ?

Madi Sousa

Numerade Educator

Problem 69

The figure shows a horizontal line $y=c$ intersecting the curve $y=8 x-27 x^{3}$. Find the number $c$ such that the areas of the shaded regions are equal.

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 70

For what values of $m$ do the line $y=m x$ and the curve $y=x /\left(x^{2}+1\right)$ enclose a region? Find the area of the region.

Linda Hand

Numerade Educator