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Calculus

James Stewart

Chapter 5

Applications of Integration - all with Video Answers

Educators

Section 1

5.1 Exercises

05:23

Problem 1

find the area of the shaded region. (Graph Can't Copy)

Noraney Ocampo
Noraney Ocampo
Numerade Educator
05:23

Problem 2

Find the area of the shaded region. (Graph Can't Copy)

Noraney Ocampo
Noraney Ocampo
Numerade Educator
05:23

Problem 3

Find the area of the shaded region. (Graph Can't Copy)

Noraney Ocampo
Noraney Ocampo
Numerade Educator
05:23

Problem 4

Find the area of the shaded region. (Graph Can't Copy)

Noraney Ocampo
Noraney Ocampo
Numerade Educator
04:17

Problem 5

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.
$y=x+1, \quad y=9-x^2, \quad x=-1, \quad x=2$

Sam Low
Sam Low
Numerade Educator
05:04

Problem 6

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.
$y=\sin x, y=x, \quad x=\pi / 2, \quad x=\pi$

SL
Sky Li
Numerade Educator
09:51

Problem 7

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
03:42

Problem 8

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

Linda Hand
Linda Hand
Numerade Educator
05:53

Problem 9

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.
$y=\sqrt{x+3}, \quad y=(x+3) / 2$

Sam Low
Sam Low
Numerade Educator
05:14

Problem 10

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.
$y=\sin x, \quad y=2 x / \pi, \quad x \geqslant 0$

Sam Low
Sam Low
Numerade Educator
06:31

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

SL
Sky Li
Numerade Educator
10:44

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

SL
Sky Li
Numerade Educator
10:16

Problem 13

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 14

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
05:25

Problem 15

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

SL
Sky Li
Numerade Educator
00:43

Problem 16

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
05:08

Problem 17

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
06:18

Problem 18

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
06:22

Problem 19

Sketch the region enclosed by the given curves and find its area.
$y=\cos \pi x, \quad y=4 x^2-1$

SL
Sky Li
Numerade Educator
06:42

Problem 20

Sketch the region enclosed by the given curves and find its area.
$x=y^4, \quad y=\sqrt{2-x}, \quad y=0$

SL
Sky Li
Numerade Educator
02:27

Problem 21

Sketch the region enclosed by the given curves and find its area.
$y=\cos x, \quad y=1-2 x / \pi$

Adalynn Griesser
Adalynn Griesser
Numerade Educator
03:20

Problem 22

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

Gregory Higby
Gregory Higby
Numerade Educator
02:27

Problem 23

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

Adalynn Griesser
Adalynn Griesser
Numerade Educator
18:46

Problem 24

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
04:31

Problem 25

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

Sam Low
Sam Low
Numerade Educator
03:11

Problem 26

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

Gregory Higby
Gregory Higby
Numerade Educator
04:13

Problem 27

Sketch the region enclosed by the given curves and find its area.
$y=1 / x^2, \quad y=x, \quad y=\frac{1}{8} x$

Kenneth Kobos
Kenneth Kobos
Numerade Educator
04:24

Problem 28

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
04:00

Problem 29

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

Adalynn Griesser
Adalynn Griesser
Numerade Educator
05:02

Problem 30

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 31

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
04:29

Problem 32

Evaluate the integral and interpret it as the area of a region. Sketch the region.
$\int_0^4|\sqrt{x+2}-x| d x$

Sam Low
Sam Low
Numerade Educator
02:39

Problem 33

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 34

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 35

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
01:22

Problem 36

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.
$y=x^2 \cos \left(x^3\right), \quad y=x^{10}$

Fuzail Shakir
Fuzail Shakir
Numerade Educator
01:22

Problem 37

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

Kenneth Kobos
Kenneth Kobos
Numerade Educator
00:56

Problem 38

Graph the region between the curves and use your calculator to compute the area correct to five decimal places.
$y=x^6, \quad y=\sqrt{2-x^4}$

Bailey Brooks
Bailey Brooks
Numerade Educator
01:35

Problem 39

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

Kenneth Kobos
Kenneth Kobos
Numerade Educator
02:42

Problem 40

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

Kenneth Kobos
Kenneth Kobos
Numerade Educator
07:29

Problem 41

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
06:42

Problem 42

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.

Kenneth Kobos
Kenneth Kobos
Numerade Educator
09:31

Problem 43

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.

Kenneth Kobos
Kenneth Kobos
Numerade Educator
03:21

Problem 44

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
Bobby Barnes
University of North Texas
06:37

Problem 45

. 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.

Lucas Everson
Lucas Everson
Numerade Educator
View

Problem 46

If the birth rate of a population is $b(t)=2200+52.3 t+0.74 t^2$ people per year and the death rate is $d(t)=1460+28.8 t$ 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
10:42

Problem 47

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.

Chris Trentman
Chris Trentman
Numerade Educator
04:39

Problem 48

The figure shows graphs of the marginal revenue function $R^{\prime}$ and the marginal cost function $C^{\prime}$ for a manufacturer. [Recall from Section 3.7 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.

Sam Low
Sam Low
Numerade Educator
04:18

Problem 49

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 50

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
04:46

Problem 51

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.

Gregory Higby
Gregory Higby
Numerade Educator
07:38

Problem 52

(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
08:40

Problem 53

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
Catherine Ross
Numerade Educator
03:24

Problem 54

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
04:55

Problem 55

Sketch the region bounded by the given curves and find the area of the region.
$y=1 / x, \quad y=1 / x^2, \quad x=2$

Sam Low
Sam Low
Numerade Educator
03:49

Problem 56

Sketch the region bounded by the given curves and find the area of the region.
$y=\sin x, \quad y=e^x, \quad x=0, \quad x=\pi / 2$

Leon Druch
Leon Druch
Numerade Educator
05:34

Problem 57

Sketch the region bounded by the given curves and find the area of the region.
$y=\tan x, \quad y=2 \sin x, \quad-\pi / 3 \leqslant x \leqslant \pi / 3$

SL
Sky Li
Numerade Educator
04:03

Problem 58

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.

Madi Sousa
Madi Sousa
Numerade Educator