428
CHAPTER 5
5
Integrals
REVIEW
CONCEPT CHECK
Answers to the Concept Check are available at StewartCalculus.com.
1. (a)Write an expression for a Riemann sum of a function f.
Explain the meaning of the notation that you use.
(b)If f sxd > 0, what is the geometric interpretation of a
Riemann sum? Illustrate with a diagram.
(c)If f sxd takes on both positive and negative values, what is
the geometric interpretation of a Riemann sum? Illustrate
with a diagram.
2. (a)Write the definition of the definite integral of a continuous
function from a to b.
(b)What is the geometric interpretation of yab f sxd dx if f sxd > 0?
(c)What is the geometric interpretation of yab f sxd dx if f sxd
takes on both positive and negative values? Illustrate with a
diagram.
5. (a) State the Net Change Theorem.
(b)If rstd is the rate at which water flows into a reservoir,
what does ytt12 rstd dt represent?
6. Suppose a particle moves back and forth along a straight line
with velocity vstd, measured in meters per second, and
accelera­tion astd.
120
(a) What is the meaning of y60
vstd dt ?
|
|
120
(b) What is the meaning of y60 vstd dt ?
(c) What is the meaning of y60 astd dt ?
120
7. (a) Explain the meaning of the indefinite integral y f sxd dx.
(b)What is the connection between the definite integral
yab f sxd dx and the indefinite integral y f sxd dx ?
3. State the Midpoint Rule.
8. Explain exactly what is meant by the statement that “differen­
tiation and integration are inverse processes.”
4. State both parts of the Fundamental Theorem of Calculus.
9. State the Substitution Rule. In practice, how do you use it?
TRUE–FALSE QUIZ
Determine whether the statement is true or false. If it is true,
explain why. If it is false, explain why or give an example that
disproves the statement.
8. If v std is the velocity at time t of a particle moving along a
line, then yab v std dt is the distance traveled during the time
period a < t < b.
1. If f and t are continuous on fa, bg, then
9. y f 9sxd [ f sxd] 4 dx − 15 [ f sxd] 5 1 C
b
a
y f f sxd 1 tsxdg dx − y f sxd dx 1 y tsxd dx
b
b
b
a
a
a
2. If f and t are continuous on fa, bg, then
S
DS
D
y f f sxd tsxdg dx − y f sxd dx y tsxd dx
b
a
b
a
b
a
3. If f is continuous on fa, bg, then
y 5f sxd dx − 5 y f sxd dx
b
b
a
a
10. If f and t are differentiable and f sxd > tsxd for a , x , b,
then f 9sxd > t9sxd for a , x , b.
11. If f and t are continuous and f sxd > tsxd for a < x < b, then
y f sxd dx > y tsxd dx
12. y sax 2 1 bx 1 cd dx − 2 y sax 2 1 cd dx
5
5
25
0
13. All continuous functions have derivatives.
14. All continuous functions have antiderivatives.
3
0
y x f sxd dx − x y f sxd dx
b
b
a
a
a
Îy
b
f sxd dx
a
6. y f sxd dx − y f szd dz
b
b
a
a
3
3
2
2
5
d
dx
Sy
b
a
D
f sxd dx − f sxd
18. y02 sx 2 x 3 d dx represents the area under the curve y − x 2 x 3
from 0 to 2.
1
22
1
5
0
17. If f is continuous on fa, bg, then
19. y
7. If f 9 is continuous on f1, 3g, then y f 9svd dv − f s3d 2 f s1d.
2
16. If y01 f sxd dx − 0, then f sxd − 0 for 0 < x < 1.
5. If f is continuous on fa, bg and f sxd > 0, then
y sf sxd dx −
b
a
15. y e x dx − y e x dx 1 y e x dx
4. If f is continuous on fa, bg, then
b
b
a
1
3
dx − 2
x4
8
20. If f has a discontinuity at 0, then y
1
21
f sxd dx does not exist.
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
CHAPTER 5 Review
429
EXERCISES
1. U
se the given graph of f to find the Riemann sum with six
subintervals. Take the sample points to be (a) left endpoints
and (b) midpoints. In each case draw a diagram and explain
what the Riemann sum represents.
7. T
he figure shows the graphs of f, f 9, and y0x f std dt. Identify
each graph, and explain your choices.
y
b
y
c
y=ƒ
2
x
a
0
2
6
x
8. Evaluate:
(a) y
1
0
2. (a) Evaluate the Riemann sum for
f sxd − x 2 2 x 0 < x < 2
with four subintervals, taking the sample points to be
right endpoints. Explain, with the aid of a diagram,
what the Riemann sum represents.
(b)Use the definition of a definite integral (with right
end­points) to calculate the value of the integral
d arctan x
se
d dx
dx
d x arctan t
y e dt
dx 0
(c)
9. T
he graph of f consists of the three line segments shown.
If tsxd − y0x f std dt, find ts4d and t9s4d.
y
2
0
0
(c)Use the Fundamental Theorem to check your answer to
part (b).
(d)Draw a diagram to explain the geometric meaning of
the integral in part (b).
3. Evaluate
y ( x 1 s1 2 x ) dx
2
0
by interpreting it in terms of areas.
4. Express
n
lim o sin x i Dx
n l ` i−1
as a definite integral on the interval f0, g and then evaluate
the integral.
11–42 Evaluate the integral, if it exists.
11. y sx 2 1 5x d dx
12. y sx 4 2 8x 1 7d dx
13. y s1 2 x 9 d dx
0
14. y s1 2 xd9 dx
su 2 2u 2
du
1
u
4
16. y (s
u 1 1) du
17. y ys y 2 1 1d5 dy
18. y y 2s1 1 y 3 dy
21
1
15. y
9
1
0
19. y
5
1
5. If y06 f sxd dx − 10 and y04 f sxd dx − 7, find y46 f sxd dx.
6. (a)Write y sx 1 2x d dx as a limit of Riemann sums, taking the sample points to be right endpoints. Use a computer algebra system to evaluate the sum and to
compute the limit.
(b)Use the Fundamental Theorem to check your answer
to part (a).
5
1
5
t
2
10. If f is the function in Exercise 9, find t 0s4d.
0
1
f
1
y sx 2 xd dx
2
d 1 arctan x
y e dx
dx 0
(b)
dt
st 2 4d2
21. y v 2 coss v 3d dv
T
0
1
0
1
2
0
20. y sins3td dt
1
0
0
22. y
1
2
0
1
21
sin x
dx
1 1 x2
23. y
y4
t 4 tan t
dt
2y4 2 1 cos t
24. y
21
z2 1 1
dz
22
z
25. y
x
dx
x2 1 1
26. y
dx
x2 1 1
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
430
CHAPTER 5
x12
27. y
sx 1 4x
2
Integrals
29. y sin t cos t dt
esx
31. y
30. y sin x cosscos xd dx
sinsln xd
32. y
dx
x
dx
sx
51–56 Find the derivative of the function.
csc 2x
dx
1 1 cot x
28. y
dx
33. y tan x lnscos xd dx
34.
3
y s1 2 x dx
35. y
x
dx
1 1 x4
36. y sinhs1 1 4xd dx
37. y
sec tan d
1 1 sec 38. y
39. y xs1 2 xd2y3 dx
40. y
41. y
42. y
3
0
|
y4
|
x 2 2 4 dx
0
|
s1 1 tan td3 sec 2t dt
54. tsxd − y
et
dt
t
56. y − y
x4
x
cos x
44.
|
3
1
y sx 1 1 dx
48. Find the area under the graph of y − sin x and above the
x-axis, between x − 0 and x − y2.
49–50 The regions A, B, and C bounded by the graph of f and the
x-axis have areas 3, 2, and 1, respectively. Evaluate the integral.
y
f
C
x
B
(b)- y
5
1
5
1
50. (a) y f sxd dx 1 y f sxd dx
4
5
1
3
4
2
1
dx
x 12
3
1
3
61. y e x cos x dx < e 2 1
1
0
60. y
y2
y4
sin x
s2
dx <
x
2
62. y x sin21x dx < y4
1
0
63. Use the Midpoint Rule with n − 6 to approximate
y03 sinsx 3 d dx. Round to four decimal places.
2
47. Find the area under the graph of y − x 2 1 5 and above the
x-axis, between x − 0 and x − 4.
49. (a) y f sxd dx
sinst 4 d dt
x3
2
1
58. y
57. y sx 2 1 3 dx
59. y x 2 cos x dx <
raph the function f sxd − cos x sin x and use the graph to
; 46. G
guess the value of the integral y02 f sxd dx. Then evaluate the
integral to confirm your guess.
0
3x11
1 2 t2
dt
1 1 t4
59–62 Use the properties of integrals to verify the inequality.
sx 2 1 dx
; 45. Use a graph to give a rough estimate of the area of the
region that lies under the curve y − x sx , 0 < x < 4. Then
find the exact area.
A
sin x
1
2x
1
; 43–44 Evaluate the indefinite integral. Illustrate and check that
your answer is reasonable by graphing both the function and its
antiderivative (take C − 0).
y s1 1 sin x dx
1
x
0
0
43.
52. Fsxd − y st 1 sin t dt
57–58 Use Property 8 of integrals to estimate the value of the
integral.
x
dx
x23
4
t2
dt
1 1 t3
53. tsxd − y cosst 2 d dt
sx
4
x
0
55. y − y
x
0
51. Fsxd − y
| f sxd | dx
(b)- y 2 f sxd dx 1 y 6 f sxd dx
3
5
1
3
64. A particle moves along a line with velocity function
vstd − t 2 2 t, where v is measured in meters per second.
Find (a) the displacement and (b) the distance traveled by
the particle during the time interval f0, 5g.
65. Let rstd be the rate at which the world’s oil is consumed,
where t is measured in years starting at t − 0 on January 1,
2000, and rstd is measured in barrels per year. What does
20
y15
rstd dt represent?
66. A radar gun was used to record the speed of a runner at the
times given in the table. Use the Midpoint Rule to estimate
the distance the runner covered during those 5 seconds.
t (s)
v smysd
t (s)
v smysd
0
0.5
1.0
1.5
2.0
2.5
0
4.67
7.34
8.86
9.73
10.22
3.0
3.5
4.0
4.5
5.0
10.51
10.67
10.76
10.81
10.81
67. A population of honeybees increased at a rate of rstd bees
per week, where the graph of r is as shown. Use the
Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
431
CHAPTER 5 Review
Midpoint Rule with six subintervals to estimate the increase
in the bee population during the first 24 weeks.
r
12000
; 71. Estimate the value of the number c such that the area
under the curve y − sinh cx between x − 0 and x − 1 is
equal to 1.
72. Suppose that the temperature in a long, thin rod placed along
the x-axis is initially Cys2ad if x < a and 0 if x . a. It
can be shown that if the heat diffusivity of the rod is k, then
the temperature of the rod at the point x at time t is
| |
8000
4000
0
Tsx, td −
4
68. Let
f sxd −
8
H
12
16
2x 2 1
2s1 2 x 2
20
24
t (weeks)
if 23 < x < 0
if 0 < x < 1
C
ye
a
a s4kt
| |
2sx2ud 2ys4ktd
0
To find the temperature distribution that results from an initial hot spot concentrated at the origin, we need to compute
lim a l 0 T sx, td. Use l’Hospital’s Rule to find this limit.
73. If f is a continuous function such that
y f std dt − sx 2 1de 1 y e f std dt
x
x
2x
1
1
Evaluate y23
f sxd dx by interpreting the integral as a difference of areas.
69. If f is continuous and y02 f sxd dx − 6, evaluate
y
y2
0
f s2 sin d cos d
70. The Fresnel function Ssxd − y0x sin ( 12 t 2) dt was introduced
in Section 5.3. Fresnel also used the function
Csxd − y cos ( 12 t 2) dt
x
0
;
in his theory of the diffraction of light waves.
(a) On what intervals is C increasing?
(b) On what intervals is C concave upward?
(c)Use a graph to solve the following equation correct to
two decimal places:
y cos ( 12 t ) dt − 0.7
x
2
0
;
(d)Plot the graphs of C and S on the same screen. How are
these graphs related?
du
2t
1
for all x, find an explicit formula for f sxd.
74. Suppose h is a function such that hs1d − 22, h9s1d − 2,
h0s1d − 3, hs2d − 6, h9s2d − 5, h0s2d − 13, and h 0 is
continuous everywhere. Evaluate y12 h0sud du.
75. If f 9 is continuous on fa, bg, show that
2 y f sxd f 9sxd dx − f f sbdg 2 2 f f sadg 2
b
a
76. Find
lim
1
hl0 h
y
21h
2
s1 1 t 3 dt
77. If f is continuous on f0, 1g, prove that
y f sxd dx − y f s1 2 xd dx
78. Evaluate
1
nl` n
lim
1
1
0
0
FS D S D S D
1
n
9
1
2
n
9
1
3
n
9
1 ∙∙∙ 1
S DG
n
n
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
9
Problems Plus
Before you look at the solution of the following example, cover it up and first try to solve
the problem yourself.
EXAMPLE Evaluate lim
x l3
S
D
x
x sin t
dt .
y
3
x23
t
SOLUTION Let’s start by having a preliminary look at the ingredients of the function.
What happens to the first factor, xysx 2 3d, when x approaches 3? The numerator
approaches 3 and the denominator approaches 0, so we have
x
x
l ` as x l 3 1 and l 2` as x l 3 2
x23
x23
The second factor approaches y3 ssin tdyt dt, which is 0. It’s not clear what happens to
the function as a whole. (One factor is becoming large while the other is becoming
small.) So how do we proceed?
One of the principles of problem solving is recognizing something familiar. Is there
a part of the function that reminds us of something we’ve seen before? Well, the
integral
3
PS Review the Principles of Problem Solving
following Chapter 1.
y
x
3
sin t
dt
t
has x as its upper limit of integration and that type of integral occurs in Part 1 of the
Fundamental Theorem of Calculus:
d x
y f std dt − f sxd
dx a
This suggests that differentiation might be involved.
Once we start thinking about differentiation, the denominator sx 2 3d reminds us of
something else that should be familiar: one of the forms of the definition of the derivative in Chapter 2 is
Fsxd 2 Fsad
xla
x2a
F9sad − lim
and with a − 3 this becomes
Fsxd 2 Fs3d
x23
F9s3d − lim
xl3
So what is the function F in our situation? Notice that if we define
Fsxd − y
x
3
sin t
dt
t
then Fs3d − 0. What about the factor x in the numerator? That’s just a red herring, so
let’s factor it out and put together the calculation:
Another approach is to use
l’Hospital’s Rule.
lim
x l3
S
sin t
dt
3
x
x sin t
t
Fsxd 2 Fs3d
dt − lim x lim
− 3 lim
y
x l3
x l3
x l3
x23 3 t
x23
x23
D
y
− 3F9s3d − 3
x
sin 3
− sin 3 (FTC1)
3
n
432
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Problems
1. If x sin x − y f std dt, where f is a continuous function, find f s4d.
x2
0
2. Suppose f is continuous, f s0d − 0, f s1d − 1, f 9sxd . 0, and y0 f sxd dx − 13. Find the value
of the integral y01 f 21s yd dy.
1
4
3. If y0 e sx22d dx − k, find the value of y0 xe sx22d dx.
4
4
4
2
3
; 4. (a)Graph several members of the family of functions f sxd − s2cx 2 x dyc for c . 0 and
look at the regions enclosed by these curves and the x-axis. Make a conjecture about
how the areas of these regions are related.
(b) Prove your conjecture in part (a).
(c)Take another look at the graphs in part (a) and use them to sketch the curve traced out
by the vertices (highest points) of the family of functions. Can you guess what kind of
curve this is?
(d) Find an equation of the curve you sketched in part (c).
5. If f sxd − y
1
tsxd
s1 1 t
0
dt, where tsxd − y
cos x
0
3
f1 1 sinst 2 dg dt, find f 9sy2d.
6. If f sxd − y0x x 2 sinst 2 d dt, find f 9sxd.
7. Evaluate lim x l 0 s1yxd y0 s1 2 tan 2td1yt dt. [Assume that the integrand is defined and continuous at t − 0; see Exercise 5.3.82.]
x
8. The figure shows two regions in the first quadrant: Astd is the area under the curve
y − sinsx 2 d from 0 to t, and Bstd is the area of the triangle with vertices O, P, and st, 0d.
Find lim1 fAstdyBstdg.
tl0
y
y
P { t, sin( t @ )} P { t, sin( t @ )}
y
y
P { t, sin( t @ ) }P { t, sin(t@ ) }
y=sin{≈} y=sin{≈}
A(t)
O
O
B(t)
A(t)
t
x
t
x
O
O
B(t)
t
x
t
x
9. Find the interval fa, bg for which the value of the integral ya s2 1 x 2 x 2 d dx is a maximum.
b
10,000
10. U
se an integral to estimate the sum o si .
i−1
11. (a)Evaluate y v xb dx, where n is a positive integer.
n
0
(b) Evaluate ya v xb dx, where a and b are real numbers with 0 < a , b.
b
12. F
ind
d2
dx 2
S
D
y y s1 1 u du dt.
x
0
sin t
1
4
13. S
uppose the coefficients of the cubic polynomial Psxd − a 1 bx 1 cx 2 1 dx 3 satisfy the
equation
b
c
d
a1 1 1 −0
2
3
4
Show that the equation Psxd − 0 has a solution between 0 and 1. Can you generalize this
result for an nth-degree polynomial?
14. A circular disk of radius r is used in an evaporator and is rotated in a vertical plane. If it is to
be partially submerged in the liquid so as to maximize the exposed wetted area of the disk,
show that the center of the disk should be positioned at a height rys1 1 2 above the surface of the liquid.
433
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
15. Prove that if f is continuous, then y f sudsx 2 ud du − y
x
x
0
0
Sy D
u
0
f std dt du.
16. The figure shows a parabolic segment, that is, a portion of a parabola cut off by a chord AB.
It also shows a point C on the parabola with the property that the tangent line at C is parallel
to the chord AB. Archimedes proved that the area of the parabolic segment is 43 times the area
of the inscribed triangle ABC. Verify Archimedes’ result for the parabola y − 4 2 x 2 and
the line y − x 1 2.
C
B
A
17. Given the point sa, bd in the first quadrant, find the downward-opening parabola that passes
through the point sa, bd and the origin such that the area under the parabola is a minimum.
18. T
he figure shows a region consisting of all points inside a square that are closer to the center
than to the sides of the square. Find the area of the region.
2
2
2
2
19. Evaluate
lim
nl`
S
1
sn sn 1 1
1
1
sn sn 1 2
1 ∙∙∙ 1
1
sn sn 1 n
D
20. For any number c, we let fc sxd be the smaller of the two numbers sx 2 cd 2 and sx 2 c 2 2d 2.
Then we define tscd − y01 fc sxd dx. Find the maximum and minimum values of tscd if
22 < c < 2.
434
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.