Blank B.E., Krantz S.G. Calculus: Single Variable

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is proportional to the amount of the drug in the bloodstream. If m(t) is the amount

of the drug in the bloodstream, then

5 α 2 β  m ð7:7:14Þ

for constants α and β.

2. According to Fick’s Law, the diffusion of a solute through a cell membrane is described by

5 k



2 cð tÞ



: ð7:7:15Þ

Here c(t) is the concentration of the solute in the cell, c

is the concentration

outside the cell, and k is a positive constant.

3. A pacemaker attached to the heart creates a potassium current I that is determined

by the differential equation

5 αð1 2 IÞ2 λI ð7:7:16Þ

involving positive constants α and λ.

4. In an evolving galaxy, the rate equation for the production and absorption of

electromagnetic radiation N has the form

5 α ð1 2 k  NÞð7:7:17Þ

where α and k are positive constants.

QUICK QUIZ

1. What is the integrating factor for the linear differential equation

y 5 x

2. If PðxÞ5

pðxÞdx, then what is the antiderivative of e

PðxÞ

1 e

PðxÞ

pðxÞy?

3. What is the general solution of

y 5 2?

4. If y 5 2 1 Ce

23t

is the general solution of

1 β t 5 α, then what are α and β?

Answers

1. 1/x

2. e

P(x)

y 1 C 3. y 5 x 1 C/x 4. α 5 6; β 5 3

EXERCISES

Problems for Practice

c In each of Exercises 1216, write the linear differential

equation in the standard form that is given by equation

(7.7.1). Determine the integrating factor e

P(x)

(in the notation

of Theorem 1) and use it, as illustrated by Example 3, to solve

the given equation. b

5 3y

y 5 0

1 4y 5 0

1 2y 5 1

5 1 1

5 15x

2xy

1 1 x

5 1 2

y 5 7

ﬃﬃﬃ

7.7 First Order Differential Equations—Linear Equations 615

1 xy 5 x

10.

1 2xy 5 expð2x

11.

2 3y 5 e

12.

y 5 e

2x=2

13.

1 y 5 6e

14.

5 cscðxÞ1 y cotð xÞ

15.

1 2xy 5 8x

16.

xlnðxÞ

5 1

c In each of Exercises 17228, solve the given initial value

problem. b

17.

1 2y 5 3 yð0Þ5 1=2

18.

1 2 5 yyð0Þ521

19. x

1 y 5 6x

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

3 1 x

yð1Þ5 7

20.

5 2 cosðπx

Þ yð2Þ521

21.

y 5 7

ﬃﬃﬃ

yð4Þ5 17

22.

1 2xy 5 expð2x

Þ yð0Þ5 5

23.

2 y 5 e

yð0Þ523

24.

1 y 5 3 1 e

yð0Þ521

25.

1 2y 5 4xyð0Þ5 3

26.

1 2x 5 yyð0Þ5 7

27.

5 y 1 x

yð0Þ5 1

28.

1 y 5 ð1 1 e

yð0Þ5 0

29. A mixing tank contains 200 L of water in which 10 kg salt is

dissolved. At time t 5 0, a valve is opened, and a solution

with1 kg saltper 10 L water entersthe tankat the rate of 20 L

per minute. An outlet pipe maintains the volume of ﬂuid in

the tank by allowing20 L of the thoroughlymixedsolutionto

ﬂowout each minute. What is the mass m(t) of salt in the tank

at time t?Whatism

5 lim

t-N

mðtÞ?

30. A mixing tank contains 100 gallons of water in which 60

pounds of salt is dissolved. At time t 5 0, a valve is

opened, and a solution with 1 pound of salt per 5 gallons

of water enters the tank at the rate of 2.5 gallons per

minute. An outlet pipe maintains the volume of ﬂuid in

the tank by allowing 2.5 gallons of the thoroughly mixed

solution to ﬂow out each minute. What is the weight m(t)

of salt in the tank at time t? What is m

5 lim

t-N

mðtÞ?

31. A mixing tank contains 100 gallons of water in which 60

pounds of salt is dissolved. At time t 5 0, a valve is opened,

and water enters the tank at the rate of 2.5 gallons per

minute. An outlet pipe maintains the volume of ﬂuid in the

tank by allowing 2.5 gallons of the thoroughly mixed

solution to ﬂow out each minute. What is the weight m(t)

of salt in the tank at time t? What is m

5 lim

t-N

mðtÞ?

32. A mixing tank contains 200 L of water in which 10 kg salt

is dissolved. At time t 5 0, a valve is opened and water

enters the tank at the rate of 20 L per minute. An outlet

pipe maintains the volume of ﬂuid in the tank by allowing

20 L of the thoroughly mixed solution to ﬂow out each

minute. What is the mass m(t) of salt in the tank at time t?

What is m

5 lim

t-N

mðtÞ?

33. A hot bar of iron with temperature 150



F is placed in a

room with constant temperature 70



F. After 4 minutes,

the temperature of the bar is 125



F. Use Newton’s Law of

Cooling to determine the temperature of the bar after 10

minutes.

34. A hot rock with initial temperature 160



F is placed in an

environment with constant temperature 90



F. The rock

cools to 150



F in 20 minutes. Use Newton’s Law of

Cooling to determine how much longer it take the rock to

cool to 100



Further Theory and Practice

35. After falling for 10 seconds, a dropped object hits the

ground at 99% of its terminal velocity. If the linear drag

coefﬁcient k is 2 kg/s, then what is the mass of the object?

36. If a mass m that is dropped falls with linear drag coefﬁ-

cient k, then at what time does the object reach half its

terminal velocity?

37. A lake has a volume of 3 3 10

cubic feet. The con-

centration of pollutants in the lake is 0.02 % by weight.

An inﬂowing river brings clean water in at the rate of

1 million cubic feet per day. It ﬂows out at the same rate

of 1 million cubic feet of water a day. How long will it be

before the pollution level is 0.001%? (A cubic foot of

water weighs 62.43 pounds.)

38. A casserole dish at temperature T

5 180



C is plunged

into a large basin of hot water whose temperature is

5 48



C. The temperature of the dish satisﬁes Newton’s

Law of Cooling. In 30 seconds, the temperature of the

dish is 80



C. How long after the immersion is it before the

temperature of the dish cools to 60



39. A thermometer is brought from a 72



F room to the out-

doors. After 1 minute, the thermometer reads 50



F, and

after a further 30 seconds, the thermometer reads 44



616 Chapter 7 Applications of the Integral

What is the actual temperature outside? (The equation to

be solved may appear to be complicated, but it can be

reduced to a quadratic equation.)

40. Suppose that R 5 30 Ω, L 5 6h,andE(t)5 12 V in Figure 3.

What is I(t)?

41. Suppose that R 5 60 Ω, L 5 12 h, and E(t) 5 24e

25t

Vin

Figure 3. What is I(t)?

42. Suppose that R 5 30 Ω, L 5 15 h, and E(t) 5 5t VinFigure3.

If I(0) 5 0, what is I(t)?

43. Suppose that R 5 15 Ω, L 5 5 h, and E(t) 5 20 sin(t)Vin

Figure 3. What is I(t)? (Use the integration formula of

Exercise 86, Section 6.1 of Chapter 6).

44. A 1000 L tank initially contains a 200 L solution in which

24 kg of salt is dissolved. Beginning at time t 5 0, an inlet

valve allows fresh water to ﬂow into the tank at the

constant rate of 12 L/min, and an outlet valve is opened so

that 10 L/min of the solution is drained. When the tank

has been ﬁlled, how much salt does it contain?

45. A 1000 L tank initially contains a 200 L solution in which

24 kg of salt is dissolved. Beginning at time t 5 0, an inlet

valve allows fresh water to ﬂow into the tank at the

constant rate of 12 L/min, and an outlet valve is opened so

that 16 L/min of the solution is drained. How much salt

does the tank contain after 25 minutes?

46. In Hull’s learning model, habit strength H(r) depends on

the number r of reinforcements according to H(r) 5

λ(1 2 e

2λr

). Show that H satisﬁes differential equation

(7.7.4). What are the constants α and β?

47. Several memory experiments have shown that when a

pattern is shown to a subject and then withdrawn, the

probability that the subject will forget the pattern in the

time interval [0, t]is

FðtÞ5 p 2 c  λ

where p, c, and λ are positive constants with c , p and

λ , 1. Show that F satisﬁes differential equation (7.7.4).

What are the constants α and β?

48. Suppose that during wartime, a particular type of weapon

is produced at a constant rate μ and destroyed in battle at

a constant relative rate δ. If there were N

of these

weapons at the outbreak of the war (t 5 0), then how

many were there at time t? If the war is a lengthy one,

about how many of these weapons will be on hand at time

t when t is large?

49. Let m(t ) be the amount of a drug in the bloodstream. If

the drug is delivered intravenously at a constant rate α

and if it is eliminated from the patient’s body at a con-

stant relative rate β, then m(t) satisﬁes equation (7.7.14).

What, approximately, is the amount of the drug that is

maintained in the bloodstream of a long-term patient?

50. Solve equation (7.7.15), which expresses Fick’s Law for

the diffusion of a solute through a cell membrane.

51. Solve equation (7.7.16) for the potassium current I ( t)

induced by a pacemaker.

52. Solve equation (7.7.17) for the electromagnetic radiation

in an evolving galaxy.

53. A tank holds a salt water solution. At time t 5 0, there are

L liters in the tank; dissolved in the L liters are m

kilo-

grams of salt. Fresh water enters the tank at a rate of ρ

liters per hour and is mixed thoroughly with the brine

solution. At the same time, the mixed solution is drained

at the rate of r liters per hour. Let m(t) denote the mass of

the salt in the tank at time t.

a. Assuming perfect mixing prior to draining, show that

m(t) satisﬁes the differential equation

ðtÞ52

L 1 ðρ 2 rÞt

mðtÞ:

b. What is m(t) in the case that ρ 5 r?

c. Supposing that ρ 6¼r, solve for m(t).

54. An object of mass m is dropped from height H. Let v

denote its velocity at time t. Assume that air drag is linear:

R(v) 52kv. Show that the height y( t) of the object at

time t is

yðtÞ5 H 2

t 1



1 2 e

2kt=m



55. Consider the object discussed in Exercise 54. Suppose

that at time τ, the value of the air drag coefﬁcient changes

from k to κ. (This happens, for example, when a skydiver

opens his parachute.) Let η(t) denote the object’s height

above Earth at time t, and let σ(t) denote the object’s

velocity at time t. For 0 # t # τ, σ(t) 5 v(t) where v(t)is

given by formula (7.7.13), and η(t) 5 y(t) where y(t)

is given by the displayed equation of Exercise 54. At time τ,

the velocity of the object is

5 vðτÞ52



1 2 e

2kτ=m



;

and its height above Earth is

5 yðτÞ5 H 2

τ 1



1 2 e

2kτ=m



The initial value problems that describe the motion after

this time τ are

dσ

52mg 2 κσ; σðτÞ5 v

and

dη

5 σ; ηðτÞ5 y

Show that for t $ τ,

σðtÞ52



1 v



exp



ðt 2 τÞ



and

ηðtÞ5 yðτÞ1

σðuÞdu:

7.7 First Order Differential Equations—Linear Equations 617

For later reference, obtain the explicit form of η(t):

ηðtÞ5 H 2

mgτ



1 2 e

2kτ=m



ðt 2 τÞ

ðmg 1 v

κÞ



1 2 e

2ðt 2 τÞκ=m



c Let k denote

a positive constant. An object is thrown

straight up from the ground with initial velocity v

. Suppose

that air resistance gives rise to the force 2kv. Let T

denote

the time for the upward trajectory, and T

the time for the

downward trajectory. Exercises 56259 outline

a derivation of

the somewhat surprising inequality T

, T

. b

56. Show

that

vðtÞ5



1 1



2kt=m

2 1



What is the value of v(T

)? Show that



1 1



57. Show that the height y(t) of the projectile at time t is

yðtÞ5



m 1





1 2 e

2kt=m



2 kt



for 0 # t # T

1 T

. In terms of the parameters of m, g,

and K, what is the height H that the projectile reaches

before beginning to fall? Use the formula for T

express y(t) in the following form:

yðtÞ5





1 2 e

2kt=m



2 kt



58. Show that the time T

for the downward trajectory is



1 1



where v

is the velocity at the time of impact with the

ground.

59. Use the second formula for y(t) found in Exercise 57,

together with the equation y(T

1 T

) 5 0, to show that

ðe

2 e

2kT

Þ5

Use Exercise 69 of Section 5.5 in Chapter 5 to deduce that

. T

. In words, it takes longer to come down than to go

up!(It may also be shown that v

. v

c Since the 1920s, mathematical models have been devel-

oped

to study running performance. These models rely on

physical theory, empirical evidence, and statistical analysis. In

1973, J. B. Keller modiﬁed an existing mathematical theory of

running to make it more appropriate for sprints. Exercises

60262 are

concerned with the Hill-Keller theory of sprinting.

In these exercises, v(t) denotes the speed of a runner at time t,

p(t) denotes the horizontal component of the propulsive force

per unit mass that is exerted by the runner at time t, and R(v)

denotes the total resistive force per unit mass. b

60. Assume

that R(v) 5 v/τ for some positive constant τ (the

dimension of which is time). Suppose also that p(t) 5 P

for a positive constant P (the unit of which is distance/

time

). Obviously, this assumption on p(t) is not tenable

for races longer than a sprint. Show that

5 P 2

Find an explicit solution for v(t). (What initial condition

must you use?)

61. Use the explicit solution for v(t) found in Exercise 60 to

show that

5 lim

t-N

vðtÞ

exists. This “terminal velocity” is called maximum velo-

city in the track community.

62. Let x(t) denote the distance a sprinter has run in time t.

Show that

xðtÞ5 v



t 2 τ



1 2 e

2t=τ





Calculator/Computer Exercises

63. A 1 kg object is dropped. After 1 second, its (downward)

speed is 9.4 m/s. Assuming that the drag is proportional to

the velocity, what is the terminal velocity of the object?

64. A 0.5 kg object is dropped. Its terminal velocity is 49 m/s.

Assuming that the drag is proportional to the velocity and

that it was dropped from a sufﬁciently great height, at

what time after being dropped is its speed 40 m/s?

65. On December 7, 1941, during the attack on Pearl Harbor,

an 800 kg bomb that was dropped from an altitude of

3170, meters struck the battleship USS Arizona and set

off a series of catastrophic explosions. The ship sank in

nine minutes with a death toll of 1177. According to an

analysis of the attack that was published in 1997, the ﬂight

of the bomb lasted 26 seconds. Assume the Linear Drag

Law R(v) 52kv.

a. What was the value of k? (Use Exercise 58. Include

the units of k in your answer.)

b. What was the theoretical terminal velocity of the

bomb.

c. What was the actual velocity of the bomb when it

struck the USS Arizona?

d. Plot the height y(t) of the bomb for 0 # t # 26.

e. What would the time of fall have been had there been

no air drag? The velocity on impact?

66. Suppose that for a skydiver, the value of k in the linear

drag law R(v) 52kv is 0.682 lb s/ft. Let P(t) denote the

percentage of terminal velocity of a 120 pound skydiver t

seconds into the jump. Plot the function P. How many

618 Chapter 7 Applications of the Integral

seconds into the jump is it when the skydiver reaches 95%

of terminal velocity? How many feet has the skydiver

fallen by this time?

67. A 0.1 kg projectile is thrown up into the air with an initial

speed of 20 m/s. Assume that R(v) 52kv where k 5 0.03

kg/s

. Approximate the maximum height H above

ground that the object reaches, the impact velocity v

with

which the projectile strikes the ground, and the times T

and T

of the upward and downward trajectories. Refer to

Exercises 5659.

68. Let y(t) and v(t) denote the height and velocity of the

object of Exercise 67. In the same coordinate plane, plot

the kinetic energy T(t) 5 mv(t)

/2, the potential energy

U(t) 5 mg y(t), and the total energy E(t) 5 T(t) 1 U( t)of

the object. Separately, plot E

(t). The behavior of E(t)

and the range of values of E

(t) reﬂect a property of

resistive forces that is called nonconservative. Explain.

69. At the 1987 World Championships in Rome, split times

for the Men’s 100 Meters competition show that Ben

Johnson (Canada) and Carl Lewis (United States) each

had maximum velocity v

equal to 11.8 m/s. (Refer to

Exercises 6062 for the Hill-Keller theory of sprinting).

Johnson’s winning time was 9.83 s, and Lewis’s second

place time was 9.93 s.

a. What were the approximate values of P and τ for

these men? According to the Hill-Keller model, can

Johnson’s victory be attributed to greater propulsive

force, lesser resistance to motion, or to both factors?

b. In the same coordinate plane, plot v(t) for both run-

ners for 0 # t # 9.83.

c. In meters, what was Johnson’s winning margin?

70. As of this writing, former president George Herbert Bush

has been the only American president to have jumped

from an airplane. During World War II, the future

president parachuted from his airplane when it was shot

down. On March 25, 1997, at the age of 72, the former

president made a recreational skydive/parachute jump. In

doing this problem, refer to Exercises 54 and 55 for

notation and formulae. Assume that the mass of the

president and his jumping gear was 7.03 lb s

/ft. Assume

linear drag: R(v) 52kv during the skydive phase of the

jump and R(v) 52kv after deployment of the parachute.

Use the value k 5 1.4 lb/s.

a. The president jumped from a height of 12,500 feet and

did not open his parachute until he reached 4,500 feet.

Compute the duration τ of this skydive. According to

the model, what was the president’s velocity v(τ) when

he opened his parachute? Plot v(t) for 0 # t # τ.

b. The total time of the president’s dive and jump was

about 9 minutes. In the absence of more precise data,

use 540 seconds as the elapsed time between jumping

from the plane and touching down on the ground. As

in Exercise 55, use σ(t) and η(t) to denote President

Bush’s velocity and height above ground for

0 # t # 540. From the equation η(540) 5 0, determine k

to three signiﬁcant digits.

c. According to this model, what was the terminal velo-

city of the president’s parachute jump? With what

velocity did he touch down?

d. Plot σ(t) and η(t) for 0 # t # 540.

71. A curling stone released at the front hog line comes to a

stop in the button, 93 ft from the line, 10 seconds later. If

the deceleration of the stone is

52kv

0:05

for some positive constant k, then what was the speed of

the stone when it was released?

Summary of Key Topics in Chapter 7

Solids of Revolution:

The Method of Disks

(Section 7.1)

Let f be

a continuous positive function on the interval [a, b]. Let R be the region

bounded above by the graph of y 5 f (x), below by the x-axis, and on the sides by

x 5 a and x 5 b. Then the volume V of the solid generated by rotating R about the

x-axis is

V 5 π

f ðxÞ

dx:

Let g be a continuous positive function on the interval [c, d]. Let R be the

region bounded on the right by the graph of x 5 g( y ), on the left by the y-axis,

above by y 5 d, and below by y 5 c. Then the volume V of the solid generated by

rotating R about the y-axis is

V 5 π

gðyÞ

dy:

Summary of Key Topics 619

Solids of Revolution:

The Method of

Washers

(Section 7.1)

Let f and g be

continuous functions with 0 # g(x) # f (x) for x in the interval [a, b].

Let R be the region bounded above by the graph of y 5 f (x ), below by the graph of

y 5 g(x), and on the sides by x 5 a and x 5 b. Then the volume V of the solid

generated by rotating R about the x-axis is

V 5 π



f ðxÞ

2 gðxÞ



dx:

Solids of Revolution:

The Method of

Cylindrical Shells

(Section 7.1)

Let f and g be

continuous functions with 0 # g(x) # f (x) for x in the interval [a, b].

Let R be the region bounded above by the graph of y 5 f (x ), below by the graph of

y 5 g(x), and on the sides by x 5 a and x 5 b.Ifc # a or b # c, then the volume V of

the solid generated by rotating R about the line x 5 c is given by

V 5 2π

jx 2 cj



f ðxÞ2 gðxÞ



dx:

Arc Length

(Section 7.2)

If f is

a continuously differentia ble function on the interval [a, b], then the arc

length of the graph of f is

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 1 f

ðxÞ

dx:

Surface Area

(Section 7.2)

If f is

a nonnegative, continuously differentiable function on the interval [a, b], then

the surface area of the surface obtained when the graph of f is rotated about the

x-axis is

2πf ðxÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 1 f

ðxÞ

dx:

Average Value

of a Function

(Section 7.3)

If f (x)

is a continuous function on an interval [a, b], then the average value of f on

[a, b]is

avg

b 2 a

f ðxÞdx:

If X is a random variable that takes values in an interval [a, b], and if f is a

nonnegative function on [a, b] such that the probability that X takes a value in any

subinterval [α, β]is

f ðxÞdx, then the mean μ

of X (or expectation of X)is

xf ðxÞdx:

Center of Mass

(Section 7.4)

If g(x) # f (x)

for x in [a, b ]; if c is any real number; if R is the region bounded above

by the graph of y 5 f (x), below by the graph of y 5 g(x), and on the sides by the line

segments x 5 a and x 5 b; and if the mass density of R is constant; then the center of

mass ð

x; yÞ of R is given by the equations

620 Chapter 7 Applications of the Integral

x 5



f ðxÞ2 gð x Þ



and

y 5



f ðxÞ

2 gðxÞ



where A is the area of R.

Work

(Section 7.5)

If a body travels a linear path with position given by x (mea

sured in feet), a # x # b,

under the inﬂuence of a force in the direction of motion which at position x is F(x),

then the work performed in traveling from a to b is

FðxÞdx:

First Order Differential

Equations

(Sections 7.6., 7.7)

A ﬁrst order differential equation is an equation of the form

5 Fðx; yÞ;

where F(x, y) is an expression that involves both x and y (in general). A differ-

entiable function y is a solution of this differential equation if y

ðxÞ5 F



x; yðxÞ



for

every x in some open interval. If F(x, y) 5 g(x)h( y), then the equation is separable,

and the (possibly implicit) solution is found by isolating the vari ables on opposite

sides of the equation and integrating:

hðyÞ

dy 5

gðxÞdx:

If F(x, y) 5 q(x) 2p(x)y, then the equation, which is said to be linear, becomes

1 pðxÞy 5 qðxÞ

and is solved by multiplying through by e

P(x)

where P(x) is an antiderivative of p(x).

The left side is then rewritten as a derivative,



PðxÞ



5 e

PðxÞ

qðxÞ;

and the solution is obtained by integrating both sides:

PðxÞ

y 5

PðxÞ

qðxÞdx 1 C:

Review Exercises For for Chapter 7

c In each of Exercises 12 20, a planar region R is described

and an axis ‘ is given. Calculate the volume V of the solid of

revolution that results when R is rotated about ‘. b

1. R is

the region bounded above by the curve y 5

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

x 2 1

below by the x-axis, and on the right by the vertical line

x 5 2; ‘ is the line y 5 0.

2. R is the region that is bounded by the x-axis, the curve

y 5

ﬃﬃﬃ

, and the line x 5 4; ‘ is the line x 5 0.

3. R is the region in the ﬁrst quadrant that is bounded

above by y 5 1 1 x

and on the right by x 5 1; ‘ is the line

y 5 0.

4. R is the region in the ﬁrst quadrant that is bounded above

by y 5 1 1 x

and on the right by x 5 1; ‘ is the line x 5 0.

Review Exercises 621

5. R is the region in the ﬁrst quadrant that is bounded above

by y 5 1 1 x

and on the right by x 5 1; ‘ is the line x 5 1.

6. R is the region in the ﬁrst quadrant that is bounded above

by y 5 1 1 x

and on the right by x 5 1; ‘ is the line y 5 2.

7. R is the region in the ﬁrst quadrant that is bounded above

by y 5 5x/(2 1 x

), below by the x-axis, and on the right by

x 5 2; ‘ is the line y 5 0.

8. R is the region in the ﬁrst quadrant that is bounded above

by y 5 5x/(2 1 x

), below by the x-axis, and on the right by

x 5 2; ‘ is the line x 5 0.

9. R is the region bounded above by y 5 4x 2 x

2 3 and

below by the x-axis; ‘ is the line x 5 0.

10. R is the ﬁrst quadrant region bounded above by y 5 2,

below by y 5 2

, and on the left by x 5 0; ‘ is the line y 5 2.

11. R is the region bounded above by y 5 1, below by y 5 1/x,

and on the right by x 5 2; ‘ is the line y 5 0.

12. R is the region bounded above by y 5 1, below by y 5 1/x,

and on the right by x 5 2; ‘ is the line y 5 1/2.

13. R is the region bounded above by y 5 1, below by y 5 1/x,

and on the right by x 5 2; ‘ is the line x 5 2.

14. R is the region bounded below by the graph of

y 5 2x

2 8x 1 7 and above by the graph of below the

graph of y 5 1; ‘ is the line x 5 0.

15. R is the region bounded above by the curve y 5 9 2 x

and below by the line y 5 x 1 7; ‘ is the line y 5 0.

16. R is the region between the curves y 5 x

1/4

and y 5 x

1/2

;

‘ is the line y 521.

17. R is the region bounded above by the graph of

y 5 x

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

lnðxÞ

for 1 # x # e, below by the x-axis, and on the

right by x 5 e; ‘ is the line y 5 0.

18. R is the region bounded above by the line y 5 3 2 x,

below by the line y 5

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

x 2 1

, and on the left by the line

x 5 1; ‘ is the line x 5 3.

19. R is the region in the ﬁrst quadrant below the graph of

y 5 1 1 sin(x), above the graph of y 5 cos(x), and to the

left of x 5 π/2; ‘ is the line y 5 0.

20. R is the region between the curve y 5 2 sin(x), π/6 # x #

5π/6

and the line y 5 1; ‘ is the line y 521.

c In Exercises 21226, calculate the arc length L of

the given

curve. b

21. y 5

ð4x 1 1Þ

3=2

0 # x # 1=4.

22. y 5



cosð2xÞ



0 # x # π=6.

23. y 5



1=2



3=2

1=4 # x # 1:

24. y 5

1=2 # x # 1.

25. y 5

5=2

21=2

1=4 # x # 1.

26. y 5

3=2

1=2

1 # x # 4:

c In Exercises 27230, calculate the area S of

the surface

obtained when the graph of the given function f is rotated

about the x-axis. b

27. f ðxÞ5 cosðxÞ 2π=2 # x # π=2

28. f ðxÞ5

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ﬃ

25 2 x

2 # x # 3

29. f ðxÞ5 4

ﬃﬃﬃ

0 # x # 5

30. f ðxÞ5 x

12x

1=2 # x # 1

c In each of Exercises 31234, calculate the length of the

given

parametric curve. b

31. x 5 t

2 1; y 5 2 2 3t

0 # t #

ﬃﬃﬃ

32. x 5 1 1 2cosð3 1 tÞ; y 5 3 2 2sinð3 1 tÞ 1 # t # 3

33. x 5 t

; y 5 3t

0 # t # 1

34. x 5

expð3tÞ ; y 5

expð2tÞ 0 # t # 1=2

c In each of Exercises 35242, ﬁnd the average value f

avg

the given function f on the given interval I. b

35. f (x) 5 4x 2 x

2 3 I 5 [1, 3]

36. f (x) 5 x

(4 2 x

)

I 5 [21, 2]

37. f (x) 5 3/(1 1 x) I 5 [1, 7]

38. f (x) 5 2

I 5 [21,3]

39. f (x) 5 x/(1 1 2x

)

I 5 [0, 2]

40. f (x) 5 6/(9 2 x

) I 5 [21, 2]

41. f (x) 5 ln(x) I 5 [1/2, 2]

42. f (x) 5 xe

I 5 [0, ln(3)]

c In each of Exercises 43250, calculate the mean of the

random

variable X that has the given probability density

function f on the given interval I. b

43. f (x) 5 sin(x) I 5 [0,π/2]

44. f ðxÞ5

ﬃﬃﬃ

I 5 ½1; 4

45. f ðxÞ5

ð4x 2 x

2 3Þ I 5 ½1; 3

46. f ðxÞ5

lnð2Þ

x11

I 5 ½21; 2

47. f ðxÞ5

ð1 2 x

21=2

I 5 ½0; 1=

ﬃﬃﬃ



48. f ðxÞ5

π 2 2

arcsinðxÞ I 5 ½0; 1

49. f ðxÞ5 lnðxÞ 1 # x # e

50. f ðxÞ5

lnð3Þð4 2 x

I 5 ½0; 1

c In each of Exercises 51256, ﬁnd the center of mass ð

x; yÞ

of the given region R, assuming that it has uniform unit mass

density. b

51. R is

bounded above by y 5 1/(1 1 x), below by the x-axis,

on the left by x 5 1, and on the right by x 5 2.

52. R is bounded above by y 5 x

3/2

, below by the x-axis, on

the left by x 5 1, and on the right by x 5 4.

622 Chapter 7 Applications of the Integral

53. R is bounded above by y 5 2 sin(x) for 0 # x # π/2, below

by the x-axis, and on the right by x 5 π/2.

54. R is bounded above by y 5 4, below by y 5

ﬃﬃﬃ

, and on the

left by the y-axis.

55. R is bounded above by y 5 4 2 x

and below by y 5 3x.

56. R is bounded above by y 5 6 2 x, below by y 5 x

, and on

the left by the y-axis.

57. An object that weighs W

pounds at the surface of the

earth weighs 15697444 W

(3962 1 y)

pounds when it

is y miles above the surface of the earth. How much work

is done lifting an object from the surface of the earth,

where it weighs 1600 pounds, straight up to a height 160

miles above Earth?

58. An object of mass m kg weighs 3.98621310

m r

when it is a distance r m from Earth’s center. How much

work is done lifting a 200 kg object straight up from the

surface of the earth to a height 300 km above the surface?

(Use 6375580 m for the radius of Earth.)

59. How high will 3 3 10

J of work lift a 100 kg mass straight

up from the surface of the earth? (Refer to the preceding

exercise for pertinent information.)

60. A man on the roof of a 30 ft high building uses a chain to

pull up a 60 lb container of tar. The chain weighs 2 pounds

per linear foot. How much work does the man do in

pulling the container from ground level to the roof?

61. A heavy uniform cable is used to lift a 200 kg load from

ground level to the top of a building that is 20 m high. If

the cable weighs 160 newtons per linear meter, then how

much work is done?

62. A heavy uniform cable weighing 20 pounds per linear foot

was used to lift a 300 pound load from ground level to the

top of a 100 foot high building. How much work was done

by the time the load was 60 ft above ground?

63. If a spring with spring constant 10 pounds per inch is

stretched 4 inches beyond its equilibrium position, then

how much work is done?

64. If a spring with spring constant 24 newtons per centimeter

is stretched 6 cm beyond its equilibrium position, then

how much work is done?

65. A spring is stretched 12 cm beyond its equilibrium posi-

tion. If the force required to maintain it in its stretched

position is 360 N, then how much work was done?

66. A spring is stretched 2 inches beyond its equilibrium

position. If the force required to maintain it in its stret-

ched position is 60 pounds, then how much more work

will be done if the spring is stretched 2 more inches?

67. A force of 864 newtons compresses a spring 12 cm from

its natural length. How much more work is done com-

pressing it a further 6 cm?

68. If 24 joules of work are done in stretching a spring 6 cm

beyond its natural length, then how much more work is

done stretching a further 4 cm?

69. The work done in extending a spring 5 cm beyond its

equilibrium position is 2 J. What force is required to

maintain the spring in that position?

70. A tank in the shape of a box has square base of side 5 m.

It is 5 m deep, contains 75 m

of water, and is being

pumped dry. The pump ﬂoats on the surface of the water

and pumps the water to the top of the tank, at which point

the water runs off. How much work is done in emptying

the tank?

71. A cylindrical tank is 2 m high, has base radius 0.3 m, and is

initially ﬁlled with water. A pump ﬂoating on the surface

pumps water to the top, at which point the water runs off.

How much work is done pumping out half the water?

72. A tank in the shape of a conical frustum is 2 m high. The

radius at the top is 1 m, and the radius at the bottom is

0.8 m. The tank initially held water 1.2 m deep at the center

of the tank. A surface pump reduced the depth of the

water at the center to 0.4 m. How much work was done?

c In Exercises 73280, ﬁnd the general solution of the given

differential

equation. b

73.

5 xy

1 y

74.

5 x

y 1 2y

75.

5 y 1 2x

76. x

5 x 2 2y

77.

5 2e

2x2y

78.

5 x 2 y=x

79.

5 y

=ð1 1 x

80.

5 y lnðxÞ

c In Exercises 81286, solve the given initial value

problem. b

81.

5 2y 1 xyð0Þ5 1=2

82.

5 2y

xyð2Þ5 1=2

83.

5 3 1 2yyð0Þ521

84.

2xy

1 1 y

yð1=2Þ5 1

85.

1 y 5 2e

yð0Þ5 5

86.

5 x 1 1 1 2y=xyð1Þ5 2

Review Exercises 623

87. A mixing tank contains 400 L of water in which 25 kg salt

is dissolved. At time t 5 0, a valve is opened, and a solu-

tion with 1/10 kg salt per liter of water enters the tank at

the rate of 32 L per minute. An outlet pipe maintains the

volume of ﬂuid in the tank by allowing 32 L of the thor-

oughly mixed solution to ﬂow out each minute. What is

the mass m(t) of salt in the tank at time t? What is

5 lim

t-N

mðtÞ?

88. A mixing tank contains 100 gallons of water in which 40

pounds of salt is dissolved. At time t 5 0, a valve is

opened, and a solution with 1 pound of salt per 8 gallons

of water enters the tank at the rate of 4 gallons per

minute. An outlet pipe maintains the volume of ﬂuid in

the tank by allowing 4 gallons of the thoroughly mixed

solution to ﬂow out each minute. What is the weight m(t)

of salt in the tank at time t ? What is m

5 lim

t-N

mðtÞ?

89. A mixing tank contains 200 gallons of water in which 100

pounds of salt is dissolved. At time t 5 0, a valve is

opened, and water enters the tank at the rate of 5 gallons

per minute. An outlet pipe maintains the volume of ﬂuid

in the tank by allowing 5 gallons of the thoroughly mixed

solution to ﬂow out each minute. What is the weight m(t)

of salt in the tank at time t? What is m

5 lim

t-N

mðtÞ?

90. A mixing tank contains 100 L of water in which 10 kg salt

is dissolved. At time t 5 0, a valve is opened, and water

enters the tank at the rate of 32 L per minute. An outlet

pipe maintains the volume of ﬂuid in the tank by allowing

32 L of the thoroughly mixed solution to ﬂow out each

minute. What is the mass m(t) of salt in the tank at time t?

What is m

5 lim

t-N

mðtÞ?

624 Chapter 7 Applications of the Integral