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Single Variable Calculus: Early Transcendentals

James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin

Chapter 6

Applications of Integration - all with Video Answers

Educators

Section 1

Areas between Curves

00:50

Problem 1

(a) Set up an integral for the area of the shaded region.
(b) Evaluate the integral to find the area.
Graph can't copy

Ishita J.
Ishita J.
Numerade Educator
00:50

Problem 2

(a) Set up an integral for the area of the shaded region.
(b) Evaluate the integral to find the area.
Graph can't copy

Ishita J.
Ishita J.
Numerade Educator
00:50

Problem 3

(a) Set up an integral for the area of the shaded region.
(b) Evaluate the integral to find the area.
Graph can't copy

Ishita J.
Ishita J.
Numerade Educator
00:50

Problem 4

(a) Set up an integral for the area of the shaded region.
(b) Evaluate the integral to find the area.
Graph can't copy

Ishita J.
Ishita J.
Numerade Educator
05:23

Problem 5

Find the area of the shaded region.
Graph can't copy

Noraney Ocampo
Noraney Ocampo
Numerade Educator
05:23

Problem 6

Find the area of the shaded region.
Graph can't copy

Noraney Ocampo
Noraney Ocampo
Numerade Educator
01:34

Problem 7

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
Gregory Higby
Numerade Educator
02:06

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
Gregory Higby
Numerade Educator
02:21

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
Gregory Higby
Numerade Educator
02:13

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
Gregory Higby
Numerade Educator
05:05

Problem 11

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
Khushbu Rani
Numerade Educator
02:31

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
Gregory Higby
Numerade Educator
05:06

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
01:25

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
09:51

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
04:42

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

Sam Low
Sam Low
Numerade Educator
06:31

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
10:44

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.
$4 x+y^2=12, \quad x=y$

SL
Sky Li
Numerade Educator
10:16

Problem 19

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

Ian Musser
Ian Musser
Numerade Educator
01:01

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
Fuzail Shakir
Numerade Educator
05:08

Problem 21

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

SL
Sky Li
Numerade Educator
06:18

Problem 22

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

Sam Low
Sam Low
Numerade Educator
03:59

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
Gregory Higby
Numerade Educator
03:20

Problem 24

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

Gregory Higby
Gregory Higby
Numerade Educator
03:42

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
Gregory Higby
Numerade Educator
00:43

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
Fuzail Shakir
Numerade Educator
02:27

Problem 27

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

Adalynn Griesser
Adalynn Griesser
Numerade Educator
18:46

Problem 28

Sketch the region enclosed by the given curves and find its area.
$y=\cos x, \quad y=1-\cos x, \quad 0 \leqslant x \leqslant \pi$

AS
Apoorva Shah
Numerade Educator
03:00

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
Sam Low
Numerade Educator
03:20

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
Gregory Higby
Numerade Educator
03:20

Problem 31

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

Gregory Higby
Gregory Higby
Numerade Educator
02:23

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
02:49

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
Gregory Higby
Numerade Educator
07:53

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
Andrija Isakov
Numerade Educator
04:13

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
Kenneth Kobos
Numerade Educator
04:24

Problem 36

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

Kenneth Kobos
Kenneth Kobos
Numerade Educator
09:15

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 $\int_0^5[f(x)-g(x)] d x$ ?
Graph can't copy

Mutahar Mehkri
Mutahar Mehkri
Numerade Educator
10:45

Problem 38

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

Kenneth Kobos
Kenneth Kobos
Numerade Educator
10:45

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
Kenneth Kobos
Numerade Educator
04:46

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
Kenneth Kobos
Numerade Educator
04:00

Problem 41

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
05:02

Problem 42

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

Kenneth Kobos
Kenneth Kobos
Numerade Educator
06:37

Problem 43

Evaluate the integral and interpret it as the area of a region. Sketch the region.
$\int_0^{\pi / 2}|\sin x-\cos 2 x| d x$

Sam Low
Sam Low
Numerade Educator
04:40

Problem 44

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

Kenneth Kobos
Kenneth Kobos
Numerade Educator
02:39

Problem 45

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.
$y=x \sin \left(x^2\right), \quad y=x^4, \quad x \geqslant 0$

Sam Low
Sam Low
Numerade Educator
03:21

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
Sam Low
Numerade Educator
06:52

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
Kenneth Kobos
Numerade Educator
02:20

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
Kenneth Kobos
Numerade Educator
00:56

Problem 49

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

Bailey Brooks
Bailey Brooks
Numerade Educator
01:16

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
01:29

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
02:51

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
07:29

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
Kenneth Kobos
Numerade Educator
05:10

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
Sam Low
Numerade Educator
09:31

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 miles 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.
Table can't copy

Kenneth Kobos
Kenneth Kobos
Numerade Educator
03:21

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.
Figure can't copy

Bobby Barnes
Bobby Barnes
University of North Texas
06:37

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.
Figure can't copy

Lucas Everson
Lucas Everson
Numerade Educator
View

Problem 58

If the birth rate of a population is $b(t)=2200 e^{0.004 t}$ 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?

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
05:22

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(b). On what day does infectiousness end?
(c) Compute the level of infectiousness for this patient.

Sam Low
Sam Low
Numerade Educator
01:59

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.

James Kiss
James Kiss
Numerade Educator
10:42

Problem 61

Two cars, A and B, start side by side and accelerate from rest. The figure shows the graphs of their velocity functions.
(a) Which car is ahead after one minute? Explain.
(b) What is the meaning of the area of the shaded region?
(c) Which car is ahead after two minutes? Explain.
(d) Estimate the time at which the cars are again side by side.
Figure can't copy

Chris Trentman
Chris Trentman
Numerade Educator
02:57

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.
Figure can't copy

James Kiss
James Kiss
Numerade Educator
04:18

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
Madi Sousa
Numerade Educator
01:06

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
Fuzail Shakir
Numerade Educator
09:10

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
Catherine Ross
Numerade Educator
07:38

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
View

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 .

Brian Wang
Brian Wang
Numerade Educator
03:24

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
Madi Sousa
Numerade Educator
06:32

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.
Figure can't copy

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
07:31

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
Linda Hand
Numerade Educator