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CLP-2 Integral Calculus

Appendix F Answers to Exercises

1 Integration
1.1 Definition of the Integral
1.1.8 Exercises

1.1.8.1.

Answer.
The area is between 1.5 and 2.5 square units.

1.1.8.2.

Answer.
The shaded area is between 2.75 and 4.25 square units. (Other estimates are possible, but this is a reasonable estimate, using methods from this chapter.)

1.1.8.3.

Answer.
The area under the curve is a number in the interval (38[12+12],38[1+12]).

1.1.8.4.

Answer.
left

1.1.8.5.

Answer.
Many answers are possible. One example is f(x)=sinx, [a,b]=[0,π], n=1. Another example is f(x)=sinx, [a,b]=[0,5π], n=5.

1.1.8.6.

Answer.
Some of the possible answers are given, but more exist.
  1. i=37i;i=15(i+2)
  2. i=372i;i=15(2i+4)
  3. i=37(2i+1);i=15(2i+5)
  4. i=18(2i1);i=07(2i+1)

1.1.8.7.

Answer.
Some answers are below, but others are possible.
  1. i=1413i and i=14(13)i
  2. i=1423i and i=142(13)i
  3. i=14(1)i23i and i=142(3)i
  4. i=14(1)i+123i and i=142(3)i

1.1.8.8.

Answer.
  1. i=152i13i
  2. i=1513i+2
  3. i=17i104i and i=17i10i4

1.1.8.9.

Answer.
  1. 52[1(35)101]
  2. 52(35)50[1(35)51]
  3. 270
  4. 1(1e)be1+e4[b(b+1)]2

1.1.8.10.

Answer.
  1. 5051=2550
  2. [12(95)(96)]2[12(4)(5)]2=20,793,500
  3. 1
  4. 10

1.1.8.11.

Answer.

1.1.8.12. (✳).

Answer.
n=4, a=2, and b=6

1.1.8.13.

Answer.
One answer is below, but other interpretations exist.

1.1.8.14.

Answer.
Many interpretations are possible--see the solution to Question 13 for a more thorough discussion--but the most obvious is given below.

1.1.8.15. (✳).

Answer.
Three answers are possible. It is a midpoint Riemann sum for f on the interval [1,5] with n=4. It is also a left Riemann sum for f on the interval [1.5,5.5] with n=4. It is also a right Riemann sum for f on the interval [0.5,4.5] with n=4.

1.1.8.16.

Answer.
252

1.1.8.17.

Answer.
212

1.1.8.18. (✳).

Answer.
i=150(5+(i12)15)8 15

1.1.8.19. (✳).

Answer.
54

1.1.8.20. (✳).

Answer.
17f(x)dx=limni=1nf(1+8in)8n

1.1.8.21. (✳).

Answer.
f(x)=sin2(2+x) and b=4

1.1.8.22. (✳).

Answer.
f(x)=x1x2

1.1.8.23. (✳).

Answer.
03ex/3cos(x)dx

1.1.8.24. (✳).

Answer.
01xexdx

1.1.8.25. (✳).

Answer.
Possible answers include:
02e1x dx13ex dx21/23/2e2x dx and201e12x dx.

1.1.8.26.

Answer.
r3n+31r31

1.1.8.27.

Answer.
r5(r961r1)

1.1.8.28. (✳).

Answer.
5

1.1.8.29.

Answer.
16

1.1.8.30.

Answer.
b2a22

1.1.8.31.

Answer.
b2a22

1.1.8.32.

Answer.
4π

1.1.8.33. (✳).

Answer.
03f(x)dx=2.5

1.1.8.34. (✳).

Answer.
53 m

1.1.8.35.

Answer.
true

1.1.8.36.

Answer.
3200 km

1.1.8.37. (✳).

Answer.
(a) There are many possible answers. Two are 204x2dx and 024(2+x)2dx.
(b) π

1.1.8.38. (✳).

Answer.
(a) 30
(b) 4114

1.1.8.39. (✳).

Answer.
563

1.1.8.40. (✳).

Answer.
6

1.1.8.41. (✳).

Answer.
12

1.1.8.42.

Answer.
f(x)=310(x5+8)2sin(2x5+2)

1.1.8.43.

Answer.
1log2

1.1.8.44.

Answer.
(a) 1log10(10b10a)
(b) 1logc(cbca); yes, it agrees.

1.1.8.45.

Answer.
π412arccos(a)+12a1a2

1.1.8.46.

Answer.
  1. [f(b)f(a)]ban
  2. Choose n to be an integer that is greater than or equal to 100[f(b)f(a)](ba).

1.1.8.47.

Answer.
true (but note, for a non-linear function, it is possible that the midpoint Riemann sum is not the average of the other two)

1.2 Basic properties of the definite integral
1.2.3 Exercises

1.2.3.1.

Answer.
Possible drawings:

1.2.3.2.

Answer.
sinbsina

1.2.3.3. (✳).

Answer.
(a) False. For example, the function
f(x)={0for x<01for x0
provides a counterexample.
(b) False. For example, the function f(x)=x provides a counterexample.
(c) False. For example, the functions
f(x)={0for x<121for x12andg(x)={0for x121for x<12
provide a counterexample.

1.2.3.4.

Answer.
(a) 120, (b) positive, (c) negative, (d) positive.

1.2.3.5. (✳).

Answer.
21

1.2.3.6. (✳).

Answer.
6

1.2.3.7. (✳).

Answer.
20

1.2.3.8.

Answer.
  1. π412arccos(a)12a1a2=π4+12arccos(a)12a1a2
  2. 12arccos(a)12a1a2

1.2.3.9. (✳).

Answer.
5

1.2.3.10.

Answer.
0

1.2.3.11.

Answer.
5

1.2.3.12. (✳).

Answer.
20+2π

1.2.3.13. (✳).

Answer.
0

1.2.3.14. (✳).

Answer.
0

1.2.3.15.

Answer.
0

1.2.3.16.

Answer.
(a) y=1b1(ax)2
(b) ab1a1a1a2x2dx
(c) πab

1.2.3.17.

Answer.
× even odd
even even odd
odd odd even

1.2.3.18.

Answer.
f(0)=0; g(0) can be any real number

1.2.3.19.

Answer.
f(x)=0 for every x

1.2.3.20.

Answer.
The derivative of an even function is odd, and the derivative of an odd function is even.

1.3 The Fundamental Theorem of Calculus
1.3.2 Exercises

1.3.2.1. (✳).

Answer.
e2e2

1.3.2.2. (✳).

Answer.
F(x)=x44+12cos2x+12.

1.3.2.3. (✳).

Answer.
(a) True
(b) False
(c) False, unless abf(x)dx=abxf(x)dx=0.

1.3.2.4.

Answer.
false

1.3.2.5.

Answer.
false

1.3.2.6.

Answer.
sin(x2)

1.3.2.7.

Answer.
e3

1.3.2.8.

Answer.
For any constant C, F(x)+C is an antiderivative of f(x). So, for example, F(x) and F(x)+1 are both antiderivatives of f(x).

1.3.2.9.

Answer.
  1. We differentiate with respect to a. Recall ddx{arccosx}=11x2. To differentiate 12a1a2, we use the product and chain rules.
    dda{π412arccos(a)+12a1a2}=01211a2+(12a)2a21a2+121a2=121a2a221a2+1a221a2=1a2+1a221a2=2(1a2)21a2=1a2
  2. F(x)=5π412arccos(x)+12x1x2

1.3.2.10.

Answer.
(a) 0
(b),(c) The FTC does not apply, because the integrand is not continuous over the interval of integration.

1.3.2.11.

Answer.

1.3.2.12.

Answer.
(a) zero
(b) increasing when 0<x<1 and 3<x<4; decreasing when 1<x<3

1.3.2.13.

Answer.
(a) zero
(b) G(x) is increasing when 1<x<3, and it is decreasing when 0<x<1 and when 3<x<4.

1.3.2.14.

Answer.
Using the definition of the derivative,
F(x)=limh0F(x+h)F(x)h=limh0ax+htdtaxtdth=limh0xx+htdth

The numerator describes the area of a trapezoid with base h and heights x and x+h.

=limh012h(x+x+h)h=limh0(x+12h)=x
So, F(x)=x.

1.3.2.15.

Answer.
f(t)=0

1.3.2.16.

Answer.
log(ax)dx=xlog(ax)x+C, where a is a given constant, and C is any constant.

1.3.2.17.

Answer.
x3exdx=ex(x33x2+6x6)+C

1.3.2.18.

Answer.
1x2+a2dx=log|x+x2+a2|+C when a is a given constant. As usual, C is an arbitrary constant.

1.3.2.19.

Answer.
xx(a+x)dx=x(a+x)alog(x+a+x)+C

1.3.2.20. (✳).

Answer.
5cos2

1.3.2.21. (✳).

Answer.
2

1.3.2.22.

Answer.
15arctan(5x)+C

1.3.2.23.

Answer.
arcsin(x2)+C

1.3.2.24.

Answer.
tanxx+C

1.3.2.25.

Answer.
34cos(2x)+C, or equivalently, 32sin2x+C

1.3.2.26.

Answer.
12x+14sin(2x)+C

1.3.2.27. (✳).

Answer.
F(π2)=log(3)
G(π2)=log(3)

1.3.2.28. (✳).

Answer.
f(x) is increasing when <x<1 and when 2<x<.

1.3.2.29. (✳).

Answer.
F(x)=sinxcos3x+6

1.3.2.30. (✳).

Answer.
4x3e(1+x4)2

1.3.2.31. (✳).

Answer.
(sin6x+8)cosx

1.3.2.32. (✳).

Answer.
F(1)=3e1

1.3.2.33. (✳).

Answer.
sinu1+cos3u

1.3.2.34. (✳).

Answer.
f(x)=2x

1.3.2.35. (✳).

Answer.
f(4)=4π

1.3.2.36. (✳).

Answer.
(a) (2x+1)ex2
(b) x=1/2

1.3.2.37. (✳).

Answer.
esinxesin(x4x3)(4x33x2)

1.3.2.38. (✳).

Answer.
2xcos(ex2)5x4cos(ex5)

1.3.2.39. (✳).

Answer.
exsin(ex)sin(x)

1.3.2.40. (✳).

Answer.
14

1.3.2.41. (✳).

Answer.
52

1.3.2.42. (✳).

Answer.
45m

1.3.2.43. (✳).

Answer.
f(x)=(22x)log(1+e2xx2) and f(x) achieves its absolute maximum at x=1, because f(x) is increasing for x<1 and decreasing for x>1.

1.3.2.44. (✳).

Answer.
The minimum is 01dt1+t4. As x runs from to , the function f(x)=0x22xdt1+t4 decreases until x reaches 1 and then increases all x>1. So the minimum is achieved for x=1. At x=1, x22x=1.

1.3.2.45. (✳).

Answer.
F achieves its maximum value at x=π.

1.3.2.46. (✳).

Answer.
2

1.3.2.47. (✳).

Answer.
log2

1.3.2.48.

Answer.
In the sketch below, open dots denote inflection points, and closed dots denote extrema.

1.3.2.49. (✳).

Answer.
(a) 3x20x3+1et3dt+3x5e(x3+1)3
(b) y=3(x+1)

1.3.2.50.

Answer.
Both students.

1.3.2.51.

Answer.
(a) 27(1cos3)
(b) x3sin(x)+3x2[1cos(x)]

1.3.2.52.

Answer.
If f(x)=0 for all x, then F(x) is even and possibly also odd.
If f(x)0 for some x, then F(x) is not even. It might be odd, and it might be neither even nor odd.
(Perhaps surprisingly, every antiderivative of an odd function is even.)

1.4 Substitution
1.4.2 Exercises

1.4.2.1.

Answer.
(a) true
(b) false

1.4.2.2.

Answer.
The reasoning is not sound: when we do a substitution, we need to take care of the differential (dx). Remember the method of substitution comes from the chain rule: there should be a function and its derivative. Here’s the way to do it:
Problem: Evaluate (2x+1)2dx.
Work: We use the substitution u=2x+1. Then du=2dx, so dx=12du:
(2x+1)2dx=u212du=16u3+C=16(2x+1)3+C

1.4.2.3.

Answer.
The problem is with the limits of integration, as in Question 1. Here’s how it ought to go:
Problem: Evaluate 1πcos(logt)tdt.
Work: We use the substitution u=logt, so du=1tdt. When t=1, we have u=log1=0 and when t=π, we have u=log(π). Then:
1πcos(logt)tdt=log1log(π)cos(u)du=0log(π)cos(u)du=sin(log(π))sin(0)=sin(log(π)).

1.4.2.4.

Answer.
This one is OK.

1.4.2.5. (✳).

Answer.
01f(u)1u2du. Because the denominator 1u2 vanishes when u=1, this is what is known as an improper integral. Improper integrals will be discussed in Section 1.12.

1.4.2.6.

Answer.
some constant C

1.4.2.7. (✳).

Answer.
12(sin(e)sin(1))

1.4.2.8. (✳).

Answer.
13

1.4.2.9. (✳).

Answer.
1300(x3+1)100+C

1.4.2.10. (✳).

Answer.
log4

1.4.2.11. (✳).

Answer.
log2

1.4.2.12. (✳).

Answer.
43

1.4.2.13. (✳).

Answer.
e61

1.4.2.14. (✳).

Answer.
13(4x2)3/2+C

1.4.2.15.

Answer.
elogx+C

1.4.2.16. (✳).

Answer.
0

1.4.2.17. (✳).

Answer.
12[cos1cos2]0.478

1.4.2.18.

Answer.
1212log2

1.4.2.19.

Answer.
12tan2θlog|secθ|+C

1.4.2.20.

Answer.
arctan(ex)+C

1.4.2.21.

Answer.
π423

1.4.2.22.

Answer.
12(log(cosx))2+C

1.4.2.23. (✳).

Answer.
12sin(1)

1.4.2.24. (✳).

Answer.
13[221]0.609

1.4.2.25.

Answer.
Using the definition of a definite integral with right Riemann sums:
ab2f(2x)dx=limni=1nΔx2f(2(a+iΔx))Δx=ban=limni=1n(ban)2f(2(a+i(ban)))=limni=1n(2b2an)f(2a+i(2b2an))2a2bf(x)dx=limni=1nΔxf(2a+iΔx)Δx=2b2an=limni=1n(2b2an)f(2a+i(2b2an))

Since the Riemann sums are exactly the same,

ab2f(2x)dx=2a2bf(x)dx

1.5 Area between curves
1.5.2 Exercises

1.5.2.1.

Answer.
Area between curves π4(2+2)

1.5.2.2.

Answer.
(a) Vertical rectangles:
(b) One possible answer:

1.5.2.3. (✳).

Answer.
02[2xx3]dx

1.5.2.4. (✳).

Answer.
3/24[45(6y2)+2y]dy

1.5.2.5. (✳).

Answer.
04a[4axx24a]dx

1.5.2.6. (✳).

Answer.
125[112(x+5)+12x]dx

1.5.2.7. (✳).

Answer.
18

1.5.2.8. (✳).

Answer.
43

1.5.2.9. (✳).

Answer.
531log2

1.5.2.10. (✳).

Answer.
8π1

1.5.2.11. (✳).

Answer.
209

1.5.2.12. (✳).

Answer.
16

1.5.2.13.

Answer.
2π

1.5.2.14. (✳).

Answer.
2[π14π2]

1.5.2.15. (✳).

Answer.
316

1.5.2.16. (✳).

Answer.
263

1.5.2.17.

Answer.
7π812

1.5.2.18.

Answer.
122134

1.6 Volumes
1.6.2 Exercises

1.6.2.1.

Answer.
The horizontal cross-sections are circles, but the vertical cross-sections are not.

1.6.2.2.

Answer.
The columns have the same volume.

1.6.2.3.

Answer.
  • Washers when 1<y6: If y>1, then our washer has inner radius 2+23y, outer radius 623y, and height dy.
  • Washers when 0y<1: When 0y<1, we have a “double washer,” two concentric rings. The inner washer has inner radius r1=y and outer radius R1=2y. The outer washer has inner radius r2=2+23y and outer radius R2=623y. The thickness of the washers is dy.

1.6.2.4. (✳).

Answer.
(a) π03xe2x2dx
(b) 01π[(3+y)2(3y)2]dy+14π[(5y)2(3y)2]dy

1.6.2.5. (✳).

Answer.
(a) 11π[(54x2)2(2x2)2]dx
(b) 10π[(5+y+1)2(5y+1)2]dy

1.6.2.6. (✳).

Answer.
π22[(9x2)2(x2+1)2]dx

1.6.2.7.

Answer.
2123

1.6.2.8. (✳).

Answer.
π4(e2a21)

1.6.2.9. (✳).

Answer.
π[38351434]=π51281

1.6.2.10. (✳).

Answer.
(a) 8π111x2dx
(b) 4π2

1.6.2.11. (✳).

Answer.
(a) The region R is the region between the blue and red curves, with 3x5, in the figures below.
(b) 43π4.19

1.6.2.12. (✳).

Answer.
(a) The region R is sketched below.
(b) π[4log232]3.998

1.6.2.13. (✳).

Answer.
π2+8π3+8π65

1.6.2.14. (✳).

Answer.
83

1.6.2.15. (✳).

Answer.
256×815=136.53˙

1.6.2.16. (✳).

Answer.
283πh

1.6.2.17.

Answer.
  • (a) 4π3b2a cubic units,
  • (b) a=16356.752 and b=16378.137,
  • (c) Approximately 1.08321×1012km3, or 1.08321×1021m3,
  • (d) Absolute error is about 3.64×109km3, and relative error is about 0.00336, or 0.336%.

1.6.2.18. (✳).

Answer.
(a) 92 (b) π12[(4x)2(1+(x1)2)2]dx

1.6.2.19. (✳).

Answer.
(a) π21 (b) π22π1.793

1.6.2.20. (✳).

Answer.
(a) V1=43πc2 (b) V2=πc3[422] (c) c=0 or c=212

1.6.2.21. (✳).

Answer.
π/2ππ[(5+πsinx)2(5+2π2x)2]dx+π3π/2π[(5+2π2x)2(5+πsinx)2] dx

1.6.2.22.

Answer.
(a) 6000cπlog2(11210), which is close to 6000cπlog2.
(b) 6km: that is, there is roughly the same mass of air in the lowest 6 km of the column as there is in the remaining 54 km.

1.7 Integration by parts
1.7.2 Exercises

1.7.2.1.

Answer.
chain; product

1.7.2.2.

Answer.
The part chosen as u will be differentiated. The part chosen as dv will be antidifferentiated.

1.7.2.3.

Answer.
f(x)g(x)dx=f(x)g(x)+f(x)g(x)g2(x)dx

1.7.2.4.

Answer.
All the antiderivatives differ only by a constant, so we can write them all as v(x)+C for some C. Then, using the formula for integration by parts,
u(x)v(x)dx=u(x)u[v(x)+C]v[v(x)+C]vu(x)dxdu=u(x)v(x)+Cu(x)v(x)u(x)dxCu(x)dx=u(x)v(x)+Cu(x)v(x)u(x)dxCu(x)+D=u(x)v(x)v(x)u(x)dx+D
where D is any constant.
Since the terms with C cancel out, it didn’t matter what we chose for C--all choices end up the same.

1.7.2.5.

Answer.
Suppose we choose dv=f(x)dx, u=1. Then v=f(x)dx, and du=dx. So, our integral becomes:
(1)uf(x)dxdv=(1)uf(x)dxv(f(x)dx)vdxdu
In order to figure out the first product (and the second integrand), you need to know the antiderivative of f(x)--but that’s exactly what you’re trying to figure out!

1.7.2.6. (✳).

Answer.
x2logx2x24+C

1.7.2.7. (✳).

Answer.
logx6x6136x6+C

1.7.2.8. (✳).

Answer.
π

1.7.2.9. (✳).

Answer.
π21

1.7.2.10.

Answer.
ex(x33x2+6x6)+C

1.7.2.11.

Answer.
x22log3x3x24log2x+3x24logx3x28+C

1.7.2.12.

Answer.
(2x2)cosx+2xsinx+C

1.7.2.13.

Answer.
(t352t2+6t)logt13t3+54t26t+C

1.7.2.14.

Answer.
es(2s4s+4)+C

1.7.2.15.

Answer.
xlog2x2xlogx+2x+C

1.7.2.16.

Answer.
ex2+1+C

1.7.2.17. (✳).

Answer.
yarccosy1y2+C

1.7.2.18. (✳).

Answer.
2y2arctan(2y)y+12arctan(2y)+C

1.7.2.19.

Answer.
x33arctanx16(1+x2)+16log(1+x2)+C

1.7.2.20.

Answer.
217ex/2cos(2x)+817ex/2sin(2x)+C

1.7.2.21.

Answer.
x2[sin(logx)cos(logx)]+C

1.7.2.22.

Answer.
2xlog2(x1log2)+C

1.7.2.23.

Answer.
2ecosx[1cosx]+C

1.7.2.24.

Answer.
xex1x+ex+C=ex1x+C

1.7.2.25. (✳).

Answer.
(a) We integrate by parts with u=sinn1x and dv=sinxdx, so that du=(n1)sinn2xcosx and v=cosx.
sinnxdx=sinn1x cosxuv+(n1)cos2x sinn2x dxvdu

Using the identity sin2x+cos2x=1,

=sinn1x cosx+(n1)(1sin2x)sinn2x dx=sinn1x cosx+(n1)sinn2x dx(n1)sinnx dx
Moving the last term on the right hand side to the left hand side gives
nsinnxdx=sinn1x cosx+(n1)sinn2x dx
Dividing across by n gives the desired reduction formula.
(b) 35256π0.4295

1.7.2.26. (✳).

Answer.
(a) Area: π4log22
(b) Volume: π22π

1.7.2.27. (✳).

Answer.
π(17e18437336)

1.7.2.28. (✳).

Answer.
12

1.7.2.29.

Answer.
2e

1.8 Trigonometric Integrals
1.8.4 Exercises

1.8.4.1.

Answer.
(e)

1.8.4.2.

Answer.
1nsecnx+C

1.8.4.3.

Answer.
We divide both sides by cos2x, and simplify.
sin2x+cos2x=1sin2x+cos2xcos2x=1cos2xsin2xcos2x+1=sec2xtan2x+1=sec2x

1.8.4.4. (✳).

Answer.
sinxsin3x3+C

1.8.4.5. (✳).

Answer.
π2

1.8.4.6. (✳).

Answer.
sin37t37sin39t39+C

1.8.4.7.

Answer.
13cos3x1cosx+C

1.8.4.8.

Answer.
π89364

1.8.4.9.

Answer.
cosx+23cos3x15cos5x+C

1.8.4.10.

Answer.
12.2sin2.2x+C

1.8.4.11.

Answer.
12tan2x+C, or equivalently, 12sec2+C

1.8.4.12. (✳).

Answer.
17sec7x15sec5x+C

1.8.4.13. (✳).

Answer.
tan49x49+tan47x47+C

1.8.4.14.

Answer.
13.5sec3.5x11.5sec1.5x+C

1.8.4.15.

Answer.
14sec4x12sec2x+C or 14tan4x+C

1.8.4.16.

Answer.
15tan5x+C

1.8.4.17.

Answer.
11.3sec1.3x+10.7cos0.7x+C

1.8.4.18.

Answer.
=14sec4xsec2x+log|secx|+C

1.8.4.19.

Answer.
41453π6

1.8.4.20.

Answer.
111+19

1.8.4.21.

Answer.
2secx+C

1.8.4.22.

Answer.
tane+1θ(tan6θ7+e+3tan4θ5+e+3tan2θ3+e+11+e)+C

1.8.4.23. (✳).

Answer.
(a) Using the trig identity tan2x=sec2x1 and the substitution y=tanx, dy=sec2x dx,
tannxdx=tann2x tan2xdx=tann2x sec2xdxtann2xdx=yn2dytann2xdx=yn1n1tann2xdx=tann1xn1tann2xdx
(b) 1315π40.0813

1.8.4.24.

Answer.
12cos2x+2log|cosx|12cos2x+C

1.8.4.25.

Answer.
tanθ+C

1.8.4.26.

Answer.
log|sinx|+C

1.8.4.27.

Answer.
12sin2(ex)+C

1.8.4.28.

Answer.
(sin2x+2)cos(cosx)+2cosxsin(cosx)+C

1.8.4.29.

Answer.
x2sin2xx4+14sinxcosx+C

1.9 Trigonometric Substitution
1.9.2 Exercises

1.9.2.1. (✳).

Answer.
(a) x=43secθ
(b) x=12sinθ
(c) x=5tanθ

1.9.2.2.

Answer.
(a) x2=3secu
(b) x1=5sinu
(c) (2x+32)=312tanu
(d) x12=12secu

1.9.2.3.

Answer.
(a) 39920
(b) 527
(c) x52

1.9.2.4.

Answer.
(a) 4x22
(b) 12
(c) 11x

1.9.2.5. (✳).

Answer.
14xx2+4+C

1.9.2.6. (✳).

Answer.
125

1.9.2.7. (✳).

Answer.
π6

1.9.2.8. (✳).

Answer.
log|1+x225+x5|+C

1.9.2.9.

Answer.
122x2+4x+C

1.9.2.10. (✳).

Answer.
116x2+16x+C

1.9.2.11. (✳).

Answer.
x299x+C

1.9.2.12. (✳).

Answer.
(a) We’ll use the trig identity cos2θ=2cos2θ1. It implies that
cos2θ=cos2θ+12cos4θ=14[cos22θ+2cos2θ+1]=14[cos4θ+12+2cos2θ+1]=cos4θ8+cos2θ2+38

So,

0π/4cos4θdθ=0π/4(cos4θ8+cos2θ2+38)dθ=[sin4θ32+sin2θ4+38θ]0π/4=14+38π4=8+3π32
as required.
(b) 8+3π16

1.9.2.13.

Answer.
0

1.9.2.14. (✳).

Answer.
2arcsinx2+x24x2+C

1.9.2.15. (✳).

Answer.
25x242arcsec5x2+C

1.9.2.16.

Answer.
403

1.9.2.17. (✳).

Answer.
arcsinx+12+C

1.9.2.18.

Answer.
14(arccos(12x3)+4x212x+8(2x3)2)+C, or equivalently, 14(arcsec(2x3)+4x212x+8(2x3)2)+C

1.9.2.19.

Answer.
log(1+2)12

1.9.2.20.

Answer.
12(arctanx+xx2+1)+C

1.9.2.21.

Answer.
3+x2x22x+2+12log|x22x+2+x1|+C

1.9.2.22.

Answer.
13log|(65x+1)+259x2+15x|+C

1.9.2.23.

Answer.
131+x2(4+x2)+log|11+x2x|+C

1.9.2.24.

Answer.
8π3+43

1.9.2.25.

Answer.
Area: 43434
Volume: π263π4

1.9.2.26.

Answer.
21+ex+2log|11+ex|x+C

1.9.2.27.

Answer.
  1. 11x2
  2. False
  3. The work in the question is not correct. The most salient problem is that when we make the substitution x=sinθ, we restrict the possible values of x to [1,1], since this is the range of the sine function. However, the original integral had no such restriction.
    How can we be sure we avoid this problem in the future? In the introductory text to Section 1.9 (before Example 1.9.1), the notes tell us that we are allowed to write our old variable as a function of a new variable (say x=s(u)) as long as that function is invertible to recover our original variable x. There is one very obvious reason why invertibility is necessary: after we antidifferentiate using our new variable u, we need to get it back in terms of our original variable, so we need to be able to recover x. Moreover, invertibility reconciles potential problems with domains: if an inverse function u=s1(x) exists, then for any x, there exists a u with s(u)=x. (This was not the case in the work for the question, because we chose x=sinθ, but if x=2, there is no corresponding θ. Note, however, that x=sinθ is invertible over [1,1], so the work is correct if we restrict x to those values.)

1.9.2.28.

Answer.
(a), (b): None.
(c): x<a

1.10 Partial Fractions
1.10.4 Exercises

1.10.4.1.

Answer.
(a) (iii)
(b) (ii)
(c) (ii)
(d) (i)

1.10.4.2. (✳).

Answer.
Ax1+B(x1)2+Cx+1+D(x+1)2+Ex+Fx2+1

1.10.4.3. (✳).

Answer.
3

1.10.4.4.

Answer.
(a) x3+2x+2x2+1=x+x+2x2+1
(b) 15x4+6x3+34x2+4x+205x2+2x+8=3x2+2+45x2+2x+8
(c) 2x5+9x3+12x2+10x+302x2+5=x3+2x+6

1.10.4.5.

Answer.
(a) 5x33x210x+6=(x+2)(x2)(5x3)
(b)
x43x25=(x+3+292)(x3+292)(x2+2932)
(c) x44x310x211x6=(x+1)(x6)(x2+x+1)
(d)
2x4+12x3x252x+15=(x+3)(x+5)(x(1+22))(x(122))

1.10.4.6.

Answer.
The goal of partial fraction decomposition is to write our integrand in a form that is easy to integrate. The antiderivative of (1) can be easily determined with the substitution u=(ax+b). It’s less clear how to find the antiderivative of (2).

1.10.4.7. (✳).

Answer.
log43

1.10.4.8. (✳).

Answer.
1xarctanx+C

1.10.4.9. (✳).

Answer.
4log|x3|2log(x2+1)+C

1.10.4.10. (✳).

Answer.
F(x)=log|x2|+log(x2+4)+2arctan(x/2)+D

1.10.4.11. (✳).

Answer.
2log|x3|+3log|x+2|+C

1.10.4.12. (✳).

Answer.
9log|x+2|+14log|x+3|+C

1.10.4.13.

Answer.
5x+12log|x1|72log|x+1|+C

1.10.4.14.

Answer.
x2x+52arctan(2x)+C

1.10.4.15.

Answer.
1x2x1+C

1.10.4.16.

Answer.
12log|x2|+12log|x+2|+32log|2x1|+C

1.10.4.17.

Answer.
log(46353)

1.10.4.18.

Answer.
12log|1cosx1+cosx|+C

1.10.4.19.

Answer.
cosx2sin2x+14log|1cosx1+cosx|+C

1.10.4.20.

Answer.
3log2+12+215(arctan(715)arctan(915))

1.10.4.21.

Answer.
=942arctan(x2)2+3x4(x2+2)+C

1.10.4.22.

Answer.
38arctanx+3x3+5x8(1+x2)2+C

1.10.4.23.

Answer.
32x2+15arctan(x5)+32log|x2+5|32x2+10+C

1.10.4.24.

Answer.
log|sinθ1sinθ2|+C

1.10.4.25.

Answer.
t12log|e2t+et+1|13arctan(2et+13)+C

1.10.4.26.

Answer.
21+ex+log|1+ex11+ex+1|+C

1.10.4.27. (✳).

Answer.
(a) The region R is
(b) 10πlog94=20πlog32
(c) 20π

1.10.4.28.

Answer.
2log53+43arctan143

1.10.4.29.

Answer.
(a) 16(log|2x3x+3|)
(b) F(x)=1x29

1.11 Numerical Integration
1.11.6 Exercises

1.11.6.1.

Answer.
Relative error: 0.08147; absolute error: 0.113; percent error: 8.147%.

1.11.6.2.

Answer.
Midpoint rule:
Trapezoidal rule:

1.11.6.3.

Answer.
M=6.25, L=2

1.11.6.4.

Answer.
One reasonable answer is M=3.

1.11.6.5.

Answer.
(a) π51808
(b) 0
(c) 0

1.11.6.6.

Answer.
Possible answers: f(x)=32x2+Cx+D for any constants C, D.

1.11.6.7.

Answer.
my mother

1.11.6.8.

Answer.
(a) true
(b) false

1.11.6.9. (✳).

Answer.
True. Because f(x) is positive and concave up, the graph of f(x) is always below the top edges of the trapezoids used in the trapezoidal rule.

1.11.6.10.

Answer.
Any polynomial of degree at most 3 will do. For example, f(x)=5x327, or f(x)=x2.

1.11.6.11.

Answer.
Midpoint:
0301x3+1dx[1(2.5)3+1+1(7.5)3+1+1(12.5)3+1+1(17.5)3+1+1(22.5)3+1+1(27.5)3+1]5
Trapezoidal:
0301x3+1dx[1/203+1+153+1+1103+1+1153+1+1203+1+1253+1+1/2303+1]5
Simpson’s:
0301x3+1dx[103+1+453+1+2103+1+4153+1+2203+1+4253+1+1303+1]53

1.11.6.12. (✳).

Answer.
2π3

1.11.6.13. (✳).

Answer.
1720π5403.5 cm3

1.11.6.14. (✳).

Answer.
π12(16.72)4.377 m3

1.11.6.15. (✳).

Answer.
12.946π0.6865 m3

1.11.6.16. (✳).

Answer.
(a) 363,500
(b) 367,000

1.11.6.17. (✳).

Answer.
(a) 492
(b) 773

1.11.6.18. (✳).

Answer.
Let f(x)=sin(x2). Then f(x)=2xcos(x2) and
f(x)=2cos(x2)4x2sin(x2).
Since |x2|1 when |x|1, and |sinθ|1 and |cosθ|1 for all θ, we have
|2cos(x2)4x2sin(x2)|2|cos(x2)|+4x2|sin(x2)|2×1+4×1×1=2+4=6
We can therefore choose M=6, and it follows that the error is at most
M[ba]324n26[1(1)]32410002=2106=2106

1.11.6.19. (✳).

Answer.
3100

1.11.6.20. (✳).

Answer.
(a)
1/33((3)5+4(133)5+2(233)5+4(2)5+2(433)5+4(533)5+(1)5)
(b) Simpson’s Rule results in a smaller error bound.

1.11.6.21. (✳).

Answer.
815

1.11.6.22. (✳).

Answer.
1180×34=114580

1.11.6.23. (✳).

Answer.
(a) T4=14[(12×1)+45+23+47+(12×12)],
(b) S4=112[1+(4×45)+(2×23)+(4×47)+12]
(c) |IS4|24180×44=11920

1.11.6.24. (✳).

Answer.
(a) T4=8.03515, S48.03509
(b)
|abf(x) dxTn|210008312(4)20.00533
|abf(x) dxSn|4100085180(4)40.00284

1.11.6.25. (✳).

Answer.
Any n68 works.

1.11.6.26. (✳).

Answer.
4723494 ft3

1.11.6.27. (✳).

Answer.
(a) 0.025635
(b) 1.8×105

1.11.6.28. (✳).

Answer.
(a) 0.6931698
(b) n12 with n even

1.11.6.29. (✳).

Answer.
(a) 0.01345
(b) n28 with n even

1.11.6.30. (✳).

Answer.
n259

1.11.6.31.

Answer.
(a) When 0x1, then x21 and x+11, so
|f(x)|=x2|x+1|11=1
(b) 12
(c) n65
(d) n46

1.11.6.32.

Answer.
x112[1+16x+3+4x+1+163x+1+1x]

1.11.6.33.

Answer.
Note: for more detail, see the solutions.
First, we use Simpson’s rule with n=4 to approximate 1211+x2dx. The choice of this method (what we’re approximating, why n=4, etc.) is explained in the solutions--here, we only show that it works.
1211+x2dx112[12+6441+813+6465+15]0.321748
For ease of notation, define A=0.321748.
Now, we bound the error associated with this approximation. Define N(x)=24(5x410x2+1) and D(x)=(x2+1)5, so N(x)/D(x) gives the fourth derivative of 11+x2. When 1x2, |N(x)|N(2)=984 (because N(x) is increasing over that interval) and |D(x)|D(1)=25 (because D(x) is also increasing over that interval), so |d4dx4{11+x2}|=|N(x)D(x)|98425=30.75. Now we find the error bound for Simpson’s rule with L=30.75, b=2, a=1, and n=4.
|1211+x2dxA|=|error|L(ba)5180n4 =30.7518044<0.00067
So,
0.00067<1211+x2dxA<0.00067A0.0067<1211+x2dx<A+0.00067A0.00067<arctan(2)arctan(1)<A+0.00067A0.00067<arctan(2)π4<A+0.00067π4+A0.00067<arctan(2)<π4+A+0.00067π4+0.3217480.00067<arctan(2)<π4+0.321748+0.00067π4+0.321078<arctan(2)<π4+0.322418π4+0.321<arctan(2)<π4+0.323
This was the desired bound.

1.12 Improper Integrals
1.12.4 Exercises

1.12.4.1.

Answer.
Any real number in [1,) or (,1], and b=±.

1.12.4.2.

Answer.
b=±

1.12.4.3.

Answer.
The red function is f(x), and the blue function is g(x).

1.12.4.4. (✳).

Answer.
False. For example, the functions f(x)=ex and g(x)=1 provide a counterexample.

1.12.4.5.

Answer.
  1. Not enough information to decide. For example, consider h(x)=0 versus h(x)=1.
  2. Not enough information to decide. For example, consider h(x)=f(x) versus h(x)=g(x).
  3. 012h(x)dx converges by the comparison test, since |h(x)|2f(x) and 02f(x)dx converges.

1.12.4.6. (✳).

Answer.
The integral diverges.

1.12.4.7. (✳).

Answer.
The integral diverges.

1.12.4.8. (✳).

Answer.
The integral does not converge.

1.12.4.9. (✳).

Answer.
The integral converges.

1.12.4.10.

Answer.
The integral diverges.

1.12.4.11.

Answer.
The integral diverges.

1.12.4.12.

Answer.
The integral diverges.

1.12.4.13.

Answer.
The integral diverges.

1.12.4.14. (✳).

Answer.
The integral diverges.

1.12.4.15. (✳).

Answer.
The integral converges.

1.12.4.16. (✳).

Answer.
The integral converges.

1.12.4.17.

Answer.
false

1.12.4.18. (✳).

Answer.
q=15

1.12.4.19.

Answer.
p>1

1.12.4.20.

Answer.
log3π4+12arctan2

1.12.4.21.

Answer.
The integral converges.

1.12.4.22.

Answer.
12

1.12.4.23. (✳).

Answer.
The integral converges.

1.12.4.24.

Answer.
The integral converges.

1.12.4.25. (✳).

Answer.
t=10 and n=2042 will do the job. There are many other correct answers.

1.12.4.26.

Answer.
(a) The integral converges.
(b) The interval converges.

1.12.4.27.

Answer.
false

1.13 More Integration Examples

Exercises

1.13.1.
Answer.
(A)–(I), (B)–(IV), (C)–(II), (D)–(III)
1.13.2.
Answer.
1527+19=8315
1.13.3.
Answer.
325arcsin(x53)+x235x2+C
1.13.4.
Answer.
0
1.13.5.
Answer.
log|x+13x+1|+C
1.13.6.
Answer.
83log279
1.13.7. (✳).
Answer.
12log|x23|+C
1.13.8. (✳).
Answer.
(a) 2
(b) 215
(c) 3e416+116
1.13.9. (✳).
Answer.
(a) 1
(b) 815
1.13.10. (✳).
Answer.
(a) e2+1
(b) log(2+1)
(c) log15130.1431
1.13.11. (✳).
Answer.
(a) 94π
(b) log22+π20.264
(c) 2log2120.886
1.13.12.
Answer.
13sin3θ2sinθ+12log|sinθ3sinθ2|+C
1.13.13. (✳).
Answer.
(a) 115
(b) 19xx2+9+C
(c) 12log|x1|14log(x2+1)12arctanx+C
(d) 12[x2arctanxx+arctanx]+C
1.13.14. (✳).
Answer.
(a) 112
(b) 2sin1x2+x1x24+C
(c) 2log|x|+1x+2log|x1|+C
1.13.15. (✳).
Answer.
(a) 25
(b) 122
(c) log2120.193
(d) log2120.193
1.13.16. (✳).
Answer.
(a) 12x2logx14x2+C
(b) 12log[x2+4x+5]3arctan(x+2)+C
(c) 12log|x3|12log|x1|+C
(d) 13arctanx3+C
1.13.17. (✳).
Answer.
(a) π412log2
(b) log|x22x+5|+12arctanx12+C
1.13.18. (✳).
Answer.
(a) 1300(x3+1)100+C
(b) sin5x5sin7x7+C
1.13.19.
Answer.
-2
1.13.20. (✳).
Answer.
(a) 14log|ex+1|+14log|ex3|+C
(b) 4π323
1.13.21. (✳).
Answer.
(a) 12sec2x+log|cosx|+C
(b) 110arctan80.1446
1.13.22.
Answer.
25(x1)5/2+23(x1)3/2+C
1.13.23.
Answer.
log|x+x22|x22x+C
1.13.24.
Answer.
724
1.13.25.
Answer.
3log|x+1|+2x+152(x+1)2+C
1.13.26.
Answer.
23arctan(23x+13)+C
1.13.27.
Answer.
12(xsinxcosx)+C
1.13.28.
Answer.
13log|x+1|16log|x2+x+1|+13arctan(2x13)+C
1.13.29.
Answer.
3x3arcsinx+31x2(1x2)3/2+C
1.13.30.
Answer.
2
1.13.31.
Answer.
14
1.13.32.
Answer.
log(log(cos(0.1))log(cos(0.2)))
1.13.33. (✳).
Answer.
(a) 12x[sin(logx)cos(logx)]+C
(b) 2log2log3=log43
1.13.34. (✳).
Answer.
(a) 94π+9
(b) 2log|x2|log(x2+4)+C
(c) π2
1.13.35.
Answer.
arcsin(1x)1xx+C
1.13.36.
Answer.
ee(e1)
1.13.37.
Answer.
exx+1+C
1.13.38.
Answer.
xsecxlog|secx+tanx|+C
1.13.39.
Answer.
x(x+a)ndx={(x+a)(n+2)n+2a(x+a)n+1n+1+C if n1,2(x+a)alog|x+a|+C if n=1log|x+a|+ax+a+C if n=2
1.13.40.
Answer.
xarctan(x2)12(12log|x22x+1x2+2x+1|+arctan(2x+1)+arctan(2x1))+C

2 Applications of Integration
2.1 Work
2.1.2 Exercises

2.1.2.1.

Answer.
0.00294 J

2.1.2.2.

Answer.
The rock has mass 19.8 kg (about 102 grams); lifting it one metre takes 1 J of work.

2.1.2.3.

Answer.
(a) metres
(b) newtons
(c) joules

2.1.2.4.

Answer.
smootbarnmegaFonzie (smoot-barns per megaFonzie)

2.1.2.5.

Answer.
10 cm below the bottom of the unloaded spring

2.1.2.6.

Answer.
x=2

2.1.2.7. (✳).

Answer.
a=3

2.1.2.8.

Answer.
(a) joules
(b) clog(11.5) J

2.1.2.9. (✳).

Answer.
14 J

2.1.2.10. (✳).

Answer.
25 J

2.1.2.11. (✳).

Answer.
196 J

2.1.2.12.

Answer.
14700 J

2.1.2.13. (✳).

Answer.
03(9.8)(8000)(2+z)(3z)2dz joules

2.1.2.14.

Answer.
0.2352 J

2.1.2.15.

Answer.
2049 kg, or about 408 grams

2.1.2.16.

Answer.
294 J

2.1.2.17.

Answer.
(a) 117.6 J
(b) 3.92[3023]104 J

2.1.2.18.

Answer.
125 m/sec, or about 22.36 cm/sec

2.1.2.19.

Answer.
yes (at least, the car won’t scrape the ground)

2.1.2.20.

Answer.
0.144 J

2.1.2.21. (✳).

Answer.
904,050π J

2.1.2.22.

Answer.
102056 J

2.1.2.23.

Answer.
(a) 4900 N
(b) 44100x2 N
(c) 29400 J

2.1.2.24.

Answer.
220.5 J

2.1.2.25.

Answer.
About 7×1028 J

2.1.2.26.

Answer.
true

2.1.2.27.

Answer.
925559 J

2.1.2.28.

Answer.
740=0.175 J

2.1.2.29.

Answer.
One possible answer: 14[1(18)4+1(38)4]

2.2 Averages
2.2.2 Exercises

2.2.2.1.

Answer.
The most straightforward of many possible answers is shown.

2.2.2.2.

Answer.
500 km

2.2.2.3.

Answer.
Wba N

2.2.2.4.

Answer.
(a) ban
(b) a+3ban
(c) f(a+3ban)
(d) 1ni=1nf(a+(i1)ban)

2.2.2.5.

Answer.
(a) yes
(b) not enough information

2.2.2.6.

Answer.
0

2.2.2.7. (✳).

Answer.
1

2.2.2.8. (✳).

Answer.
1e1[29e3+19]

2.2.2.9. (✳).

Answer.
4π+1

2.2.2.10. (✳).

Answer.
2π

2.2.2.11. (✳).

Answer.
103log7 degrees Celsius

2.2.2.12. (✳).

Answer.
12(e1)

2.2.2.13. (✳).

Answer.
12

2.2.2.14.

Answer.
(a) 400 ppm
(b) 599.99 ppm
(c) 0.125, or 12.5%

2.2.2.15.

Answer.
(a) 16π5
(b) 32π5
(c) 32π5

2.2.2.16.

Answer.
(a) 0
(b) 3

2.2.2.17.

Answer.
4π10.52

2.2.2.18.

Answer.
(a) F(t)=3f(t)=3sin(tπ) N
(b) 0
(c) 322.12

2.2.2.19. (✳).

Answer.
(a) 130 km
(b) 65 km/hr

2.2.2.20.

Answer.
(a) A=e1
(b) 0
(c) 42e+2(e1)log(e1)0.42

2.2.2.21.

Answer.
(a) neither — both are zero
(b) |f(x)A| has the larger average on [0,4]

2.2.2.22.

Answer.
(ba)πR2

2.2.2.23.

Answer.
0

2.2.2.24.

Answer.
Yes, but if a0, then s=t.

2.2.2.25.

Answer.
A

2.2.2.26.

Answer.
(a) bA(b)aA(a)ba
(b) f(t)=A(t)+tA(t)

2.2.2.27.

Answer.
  1. One of many possible answers: f(x)={1 if x01 if x>0.
  2. No such function exists.
    • Note 1: Suppose f(x)>0 for all x in [1,1]. Then 1211f(x)dx>12110dx=0. That is, the average value of f(x) on the interval [1,1] is not zero — it’s something greater than zero.
    • Note 2: Suppose f(x)<0 for all x in [1,1]. Then 1211f(x)dx<12110dx=0. That is, the average value of f(x) on the interval [1,1] is not zero — it’s something less than zero.
    So, if the average value of f(x) is zero, then f(x)0 for some x in [1,1], and f(y)0 for some y[1,1]. Since f is a continuous function, and 0 is between f(x) and f(y), by the intermediate value theorem (see the CLP-1 text) there is some value c between x and y such that f(c)=0. Since x and y are both in [1,1], then c is as well. Therefore, no function exists as described in the question.

2.2.2.28.

Answer.
true

2.2.2.29.

Answer.
0

2.3 Centre of Mass and Torque
2.3.3 Exercises

2.3.3.1.

Answer.
(1,1)

2.3.3.2.

Answer.
(0,0)

2.3.3.3.

Answer.
In general, false.

2.3.3.4.

Answer.
3.5 metres from the left end

2.3.3.5.

Answer.
(a) to the left
(b) to the left
(c) not enough information
(d) along the line x=a
(e) to the right

2.3.3.6.

Answer.
39200π9(12π)121,212 J

2.3.3.7.

Answer.
(a), (b) 1xdx
(c), (d) log3
(e), (f) 2log3

2.3.3.8.

Answer.
(a)
i=1n[banρ(a+(i12)(ban))×(a+(i12)(ban))]i=1nbanρ(a+(i12)(ban))
(b) x¯=abxρ(x)dxabρ(x)dx

2.3.3.9.

Answer.
(a)
(b) (T(x)B(x))dx
(c) T(x)B(x)
(d) x¯=abx(T(x)B(x))dxab(T(x)B(x))dx

2.3.3.10.

Answer.
(a) The strips between x=a and x=a at the left end of the figure all have the same centre of mass, which is the y-value where T(x)=B(x), x<0. So, there should be multiple weights of different mass piled up at that y-value.
Similarly, the strips between x=b and x=b at the right end of the figure all have the same centre of mass, which is the y-value where T(x)=B(x), x>0. So, there should be a second pile of weights of different mass, at that (higher) y-value.
Between these two piles, there are a collection of weights with identical mass distributed fairly evenly. The top and bottom ends of R (above the uppermost pile, and below the lowermost pile) have no weights.
One possible answer (using twelve slices):
(b) The area of the strip is (T(x)B(x))dx, and its centre of mass is at height T(x)+B(x)2.
(c) y¯=ab(T(x)2B(x)2)dx2ab(T(x)B(x))dx

2.3.3.11. (✳).

Answer.
x¯=13106x2dx

2.3.3.12.

Answer.
x¯=143

2.3.3.13.

Answer.
x¯=log10.12(arctan10+arctan(3))0.43

2.3.3.14. (✳).

Answer.
y¯=34ee4

2.3.3.15. (✳).

Answer.
(a)
(b) 3log38π

2.3.3.16. (✳).

Answer.
x¯=π42121 and y¯=14(21)

2.3.3.17. (✳).

Answer.
(a) x¯=kA[21],y¯=k2π8A
(b) k=8π[21]

2.3.3.18. (✳).

Answer.
(a)
(b) 83
(c) 1

2.3.3.19. (✳).

Answer.
2πlog20.44127

2.3.3.20. (✳).

Answer.
x¯=0 and y¯=1224+9π

2.3.3.21. (✳).

Answer.
(a) 94π
(b) x¯=0 and y¯=4π

2.3.3.22.

Answer.
(x¯,y¯)=(1,2π)

2.3.3.23.

Answer.
(e23/2e25/2,e474e210)(1.2,2.4)

2.3.3.24. (✳).

Answer.
y¯=85

2.3.3.25. (✳).

Answer.
(a) x¯=811, y¯=16655
(b) π04ydy+π46(6y)2dy

2.3.3.26. (✳).

Answer.
(a) y¯=e434e
(b) π(e22+2e32)

2.3.3.27.

Answer.
(3,1.5)

2.3.3.28.

Answer.
(0,3.45)

2.3.3.29.

Answer.
(a) h4
(b) 12h2k23hk2+14k3h2hk+13k2

2.3.3.30.

Answer.
about 0.833 N

2.3.3.31.

Answer.
(a) 17,150π J
(b) 24509π(8π9)13,797 J
(c) about 74%

2.3.3.32.

Answer.
x¯=π162π2[sin(π72)+2sin(π18)+9sin(π8)+8sin(2π9)+25sin(25π72)+9]0.976

2.4 Separable Differential Equations
2.4.7 Exercises

2.4.7.1.

Answer.
(a) yes
(b) yes
(c) no

2.4.7.2.

Answer.
  1. One possible answer: f(x)=x, g(y)=siny3y.
  2. One possible answer: f(x)=ex, g(y)=ey.
  3. One possible answer: f(x)=x1, g(y)=1.
  4. The given equation is equivalent to the equation dydx=x, which fits the form of a separable equation with f(x)=x, g(y)=1.

2.4.7.3.

Answer.
The mnemonic allows us to skip from the separable differential equation we want to solve (very first line) to the equation
1g(y)dy=f(x)dx

2.4.7.4.

Answer.
false

2.4.7.5.

Answer.
(a) [0,)
(b) No such function exists. If |f(x)|=Cx and f(x) switches from f(x)=Cx to f(x)=Cx at some point, then that point is a jump discontinuity. Where f(x) contains a discontinuity, dydx does not exist.

2.4.7.6.

Answer.
dQdt=0.003Q(t)

2.4.7.7.

Answer.
dpdt=αp(t)(1p(t)), for some constant α.

2.4.7.8.

Answer.
(a) 1
(b) 0
(c) 0.5
(d) Two possible answers are shown below:
Another possible answer is the constant function y=2.

2.4.7.9.

Answer.
(a) 12
(b) 32
(c) 52
(d) Your sketch should look something like this:
(e) There are lots of possible answers. Several are shown below.

2.4.7.10. (✳).

Answer.
y=log(x2+2)

2.4.7.11. (✳).

Answer.
y(x)=31+x2

2.4.7.12. (✳).

Answer.
y(t)=3log(3C+sint)

2.4.7.13. (✳).

Answer.
y=32ex2+C3.

2.4.7.14. (✳).

Answer.
y=log(Cx22)
The solution only exists for Cx22>0, i.e. C>0 and the function has domain {x:|x|<2C}.

2.4.7.15. (✳).

Answer.
y=(3ex3x2+24)1/3

2.4.7.16. (✳).

Answer.
y=f(x)=1x2+16

2.4.7.17. (✳).

Answer.
y=10x3+4x2+6x4

2.4.7.18. (✳).

Answer.
y(x)=ex4/4

2.4.7.19. (✳).

Answer.
y=112x

2.4.7.20. (✳).

Answer.
f(x)=eex2/2

2.4.7.21. (✳).

Answer.
y(x)=4+2log2xx+1. Note that, to satisfy y(1)=2, we need the positive square root.

2.4.7.22. (✳).

Answer.
y2+23(y24)3/2=2secx+2

2.4.7.23. (✳).

Answer.
12 weeks

2.4.7.24. (✳).

Answer.
t=mkgarctan(kmgv0)

2.4.7.25. (✳).

Answer.
(a) k=1400
(b) t=70sec

2.4.7.26. (✳).

Answer.
(a) x(t)=34ekt12ekt
(b) As t, x2.

2.4.7.27. (✳).

Answer.
(a) P=41+e4t
(b) At t=12, P3.523. As t, P4.

2.4.7.28. (✳).

Answer.
(a) dvdt=kv2
(b) v=400t+1
(c) t=7

2.4.7.29. (✳).

Answer.
(a) B(t)=Ce0.06t0.02cost with the arbitrary constant C0.
(b) $1159.89

2.4.7.30. (✳).

Answer.
(a) B(t)={3000050m}et/50+50m
(b) $600

2.4.7.31. (✳).

Answer.
y(x)=4e1cosx2e1cosx. The largest allowed interval is
arccos(1log2)<x<arccos(1log2)
or, roughly, 1.259<x<1.259.

2.4.7.32. (✳).

Answer.
180,0003g99,591 sec27.66 hr

2.4.7.33. (✳).

Answer.
t=4×144151252g2,394sec 0.665hr

2.4.7.34. (✳).

Answer.
(a) 3
(b) y=(y1)(y2)
(c) f(x)=4ex2ex

2.4.7.35. (✳).

Answer.
p=14

2.4.7.36.

Answer.
  1. One possible answer: f(t)=0
  2. 1xa[f(x)12(xa)axf(t)dt]=f2(x)2axf2(t)dt
  3. 2xaaxf(t)dt[f(x)12(xa)axf(t)dt]=f2(x)
  4. Y(x)=D(xa), where D is any constant
  5. f(t)=D, for any nonnegative constant D

2.4.7.37.

Answer.
x=14(y1+14log|2y12y+1|)

3 Sequences and series
3.1 Sequences
3.1.2 Exercises

3.1.2.1.

Answer.
(a) 2(b) 0 (c) the limit does not exist

3.1.2.2.

Answer.
true

3.1.2.3.

Answer.
(a) ABC (b) 0 (c) AB

3.1.2.4.

Answer.
Two possible answers, of many:
  • an={3000n if n10002+1n if n>1000
  • an=1,002,001n2

3.1.2.5.

Answer.
One possible answer is an=(1)n={1,1,1,1,1,1,1,}.
Another is an=n(1)n={1,2,3,4,5,6,7,}.

3.1.2.6.

Answer.
One sequence of many possible is an=(1)nn={1, 12, 13, 14, 15, 16, }.

3.1.2.7.

Answer.
Some possible answers:
  1. 1nsinnn1n
  2. n213enn2en(7+sinn5cosn)n2en or 0n2en(7+sinn5cosn)n2en
  3. 1nn(n)n1nn

3.1.2.8.

Answer.
(a) an=bn=h(n)=i(n), cn=j(n), dn=f(n), en=g(n)
(b) limnan=limnbn=limxh(x)=1, limncn=limnen=limxg(x)=limxj(x)=0, limndn, limxf(x) and limxi(x) do not exist.

3.1.2.9.

Answer.
(a) Some possible answers: a220.99996, a660.99965, and a1100.99902.
(b) Some possible answers: a110.0044, a330.0133, and a550.0221.
The integers 11, 33, and 55 were found by approximating π by 227 and finding when an odd multiple of 117 (which is the corresponding approximation of π/2) is an integer.
(c) Some possible answers: a440.9998, a1320.9986, and a2200.09961.
See the solution for how we found them.

3.1.2.10.

Answer.
(a) (b) 34 (c) 0

3.1.2.11.

Answer.

3.1.2.12.

Answer.
0

3.1.2.13.

Answer.
0

3.1.2.14.

Answer.
0

3.1.2.15.

Answer.
1

3.1.2.16.

Answer.
0

3.1.2.17.

Answer.

3.1.2.18. (✳).

Answer.
limkak=0.

3.1.2.19. (✳).

Answer.
The sequence converges to 0.

3.1.2.20. (✳).

Answer.
9

3.1.2.21. (✳).

Answer.
log2

3.1.2.22.

Answer.
5

3.1.2.23.

Answer.

3.1.2.24.

Answer.
100299.

3.1.2.25.

Answer.
Possible answers are {an}={n[f(a+1n)f(a)]} or {an}={n[f(a)f(a1n)]}.

3.1.2.26.

Answer.
(a) An=n2sin(2πn)(b) π

3.1.2.27.

Answer.
  1. An=1 for all n
  2. limnAn=1.
  3. g(x)=0
  4. 0g(x)dx=0.

3.1.2.28.

Answer.
e3

3.1.2.29.

Answer.
(a) 4(b) x=4 (c) see solution

3.1.2.30.

Answer.
(a) decreasing (b)fn=1nf1 (c) 2% (d) 0.18%
(e) “be”: 11,019,308; “and”: 7,346,205

3.2 Series
3.2.2 Exercises

3.2.2.1.

Answer.
NSN1121+1231+12+1341+12+13+1451+12+13+14+15

3.2.2.2.

Answer.
3

3.2.2.3.

Answer.
(a) an={12 if n=11n(n+1) else  (b) 0 (c) 1

3.2.2.4.

Answer.
an={0 if n=12(1)n1n(n1) else

3.2.2.5.

Answer.
an<0 for all n2

3.2.2.6.

Answer.
(a) n=124n (b) 23

3.2.2.7.

Answer.
(a) n=119n (b) 18

3.2.2.8.

Answer.
Two possible pictures:

3.2.2.9.

Answer.
5101145100

3.2.2.10.

Answer.
All together, there were 36 cookies brought by Student 11 through Student 20.

3.2.2.11.

Answer.
551145100

3.2.2.12.

Answer.
(a) As time passes, your gains increase, approaching $1. (b) 1
(c) As time passes, you lose more and more money, without bound. (d)

3.2.2.13.

Answer.
A+B+Cc1

3.2.2.14.

Answer.
in general, false

3.2.2.15. (✳).

Answer.
32

3.2.2.16. (✳).

Answer.
17×86

3.2.2.17. (✳).

Answer.
6

3.2.2.18. (✳).

Answer.
cos(π3)cos(0)=12

3.2.2.19. (✳).

Answer.
(a) an=1116n2+24n+5 (b) 34

3.2.2.20. (✳).

Answer.
245

3.2.2.21. (✳).

Answer.
730

3.2.2.22. (✳).

Answer.
26399

3.2.2.23. (✳).

Answer.
321999=107333

3.2.2.24. (✳).

Answer.
3

3.2.2.25. (✳).

Answer.
12+57=1714

3.2.2.26. (✳).

Answer.
403

3.2.2.27.

Answer.
The series diverges to .

3.2.2.28.

Answer.
12

3.2.2.29.

Answer.
9.8 J

3.2.2.30.

Answer.
4π3(π31)

3.2.2.31.

Answer.
sin238+3232.0025

3.2.2.32.

Answer.
an={2n(n1)(n2) if n3,52 if n=2,2 if n=1

3.2.2.33.

Answer.
58

3.3 Convergence Tests
3.3.11 Exercises

3.3.11.1.

Answer.
(B), (C)

3.3.11.2.

Answer.
(A)

3.3.11.3.

Answer.
(a) I am old (b) not enough information to tell
(c) not enough information to tell (d) I am young

3.3.11.4.

Answer.
if an converges if an diverges
and if {an} is the red series then bn CONVERGES inconclusive
and if {an} is the blue series inconclusive then bn DIVERGES

3.3.11.5.

Answer.
(a) both direct comparison and limit comparison
(b) direct comparison
(c) limit comparison (d) neither

3.3.11.6.

Answer.
It diverges by the divergence test, because limnan0.

3.3.11.7.

Answer.
We cannot use the divergence test to show that a series converges. It is inconclusive in this case.

3.3.11.8.

Answer.
The integral test does not apply because f(x) is not decreasing.

3.3.11.9.

Answer.
The inequality goes the wrong way, so the direct comparison test (with this comparison series) is inconclusive.

3.3.11.10.

Answer.
(B), (D)

3.3.11.11.

Answer.
One possible answer: n=11n2.

3.3.11.12.

Answer.
By the divergence test, for a series an to converge, we need limnan=0. That is, the magnitude (absolute value) of the terms needs to be getting smaller. If limn|anan+1|<1 or (equivalently) limn|an+1an|>1, then |an+1|>|an| for sufficiently large n, so the terms are actually growing in magnitude. That means the series diverges, by the divergence test.

3.3.11.13.

Answer.
One possible answer: f(x)=sin(πx), an=0 for every n.
By the integral test, any answer will use a function f(x) that is not both positive and decreasing.

3.3.11.14. (✳).

Answer.
One possible answer: bn=2n3n

3.3.11.15. (✳).

Answer.
(a) In general false. The harmonic series n=11n provides a counterexample.
(b) In general false. If an=(1)n1n, then n=1(1)n(an is again the harmonic series n=11n, which diverges.
(c) In general false. Take, for example, an=0 and bn=1.

3.3.11.16. (✳).

Answer.
No. It diverges.

3.3.11.17. (✳).

Answer.
It diverges.

3.3.11.18. (✳).

Answer.
The series diverges.

3.3.11.19.

Answer.
It diverges.

3.3.11.20.

Answer.
This is a geometric series with r=1.001. Since |r|>1, it is divergent.

3.3.11.21.

Answer.
The series converges to 1150.

3.3.11.22.

Answer.
The series converges.

3.3.11.23.

Answer.
It diverges.

3.3.11.24.

Answer.
The series converges.

3.3.11.25.

Answer.
The series converges to 13.

3.3.11.26.

Answer.
The series converges.

3.3.11.27.

Answer.
It converges.

3.3.11.28. (✳).

Answer.
Let f(x)=5x(logx)3/2. Then f(x) is positive and decreases as x increases. So the sum 3f(n) and the integral 3f(x)dx either both converge or both diverge, by the integral test, which is Theorem 3.3.5. For the integral, we use the substitution u=logx, du=dxx to get
35dxx(logx)3/2=log35duu3/2
which converges by the p-test (which is Example 1.12.8) with p=32>1.

3.3.11.29. (✳).

Answer.
p>1

3.3.11.30. (✳).

Answer.
It converges.

3.3.11.31. (✳).

Answer.
The series n=23n2 converges by the p-test with p=2.
Note that
0<an=3n27n3<3n2n3=3n2
for all n2. As the series n=23n2 converges, the comparison test says that n=23n27n3 converges too.

3.3.11.32. (✳).

Answer.
The series converges.

3.3.11.33. (✳).

Answer.
It diverges.

3.3.11.34. (✳).

Answer.
(a) diverges (b) converges

3.3.11.35. (✳).

Answer.
The series diverges.

3.3.11.36. (✳).

Answer.
(a) converges (b) diverges

3.3.11.37.

Answer.
1e5e4

3.3.11.38. (✳).

Answer.
17

3.3.11.39. (✳).

Answer.
(a) diverges by limit comparison with the harmonic series
(b) converges by the ratio test

3.3.11.40. (✳).

Answer.
(a) Converges by the limit comparison test with b=1k5/3.
(b) Diverges by the ratio test.
(c) Diverges by the integral test.

3.3.11.41. (✳).

Answer.
It converges.

3.3.11.42. (✳).

Answer.
N=5

3.3.11.43. (✳).

Answer.
N999

3.3.11.44. (✳).

Answer.
We need N=4 and then S4=132152+172192

3.3.11.45. (✳).

Answer.
(a) converges (b) converges

3.3.11.46. (✳).

Answer.
(a) See the solution.
(b) f(x)=x+sinx1+x2 is not a decreasing function.
(c) See the solution.

3.3.11.47. (✳).

Answer.
The sum is between 0.9035 and 0.9535.

3.3.11.48. (✳).

Answer.
Since limnan=0, there must be some integer N such that 12>an0 for all n>N. Then, for n>N,
an1anan11/2=2an

From the information in the problem statement, we know

n=N+12an=2n=N+1anconverges. 

So, by the direct comparison test,

n=N+1an1anconverges as well. 

Since the convergence of a series is not affected by its first N terms, as long as N is finite, we conclude

n=1an1anconverges.

3.3.11.49. (✳).

Answer.
It diverges.

3.3.11.50. (✳).

Answer.
It converges to log2=log12,

3.3.11.51. (✳).

Answer.
See the solution.

3.3.11.52.

Answer.
About 9% to 10%

3.3.11.53.

Answer.
The total population is between 29,820,091 and 30,631,021 people.

3.4 Absolute and Conditional Convergence
3.4.3 Exercises

3.4.3.1. (✳).

Answer.
False. For example, bn=1n provides a counterexample.

3.4.3.2.

Answer.
an converges an diverges
|an| converges converges absolutely not possible
|an| diverges converges conditionally diverges

3.4.3.3. (✳).

Answer.
Conditionally convergent

3.4.3.4. (✳).

Answer.
The series diverges.

3.4.3.5. (✳).

Answer.
It diverges.

3.4.3.6. (✳).

Answer.
It converges absolutely.

3.4.3.7. (✳).

Answer.
It converges absolutely.

3.4.3.8. (✳).

Answer.
It diverges.

3.4.3.9. (✳).

Answer.
It converges absolutely.

3.4.3.10.

Answer.
See solution.

3.4.3.11.

Answer.
See solution.

3.4.3.12.

Answer.
See solution.

3.4.3.13. (✳).

Answer.
(a) See the solution.
(b) |SS5|24×36e63

3.4.3.14.

Answer.
cos1389720; the actual associated error (using a calculator) is about 0.000025.

3.4.3.15.

Answer.
See solution.

3.5 Power Series
3.5.3 Exercises

3.5.3.1.

Answer.
2

3.5.3.2.

Answer.
f(x)=n=1n(x5)n1n!+2

3.5.3.3.

Answer.
Only x=c

3.5.3.4.

Answer.
R=6

3.5.3.5. (✳).

Answer.
(a) R=12
(b) 21+2x for all |x|<12

3.5.3.6. (✳).

Answer.
R=

3.5.3.7. (✳).

Answer.
1

3.5.3.8. (✳).

Answer.
The interval of convergence is 1<x+21 or (3,1].

3.5.3.9. (✳).

Answer.
The radius of convergence is 3. The interval of convergence is 4<x2, or simply (4,2].

3.5.3.10. (✳).

Answer.
3x<7 or [3,7)

3.5.3.11. (✳).

Answer.
The given series converges if and only if 3x1. Equivalently, the series has interval of convergence [3,1].

3.5.3.12. (✳).

Answer.
(a) 34x<54 or [34,54)
(b) 58x<38 or [58,38)
(c) 34<x<14, or (34,14)

3.5.3.13. (✳).

Answer.
The radius of convergence is 2. The interval of convergence is 1<x3, or (1,3].

3.5.3.14. (✳).

Answer.
The interval of convergence is   a1<x<a+1, or (a1,a+1).

3.5.3.15. (✳).

Answer.
(a) |x+1|9 or 10x8 or [10,8]
(b) This series converges only for x=1.

3.5.3.16. (✳).

Answer.
n=0xn+3=n=3xn

3.5.3.17.

Answer.
f(x)=3+n=1(x1)nn(n+1)

3.5.3.18. (✳).

Answer.
The series converges absolutely for |x|<9, converges conditionally for x=9 and diverges otherwise.

3.5.3.19. (✳).

Answer.
(a) n=0(1)nx3n+13n+1+C
(b) We need to keep two terms (the n=0 and n=1 terms).

3.5.3.20. (✳).

Answer.
(a) See the solution.
(b) n=0n2xn=x(1+x)(1x)3. The series converges for 1<x<1.

3.5.3.21. (✳).

Answer.
See the solution.

3.5.3.22. (✳).

Answer.
(a) 1.
(b) The series converges for 1x<1, i.e. for the interval [1,1)

3.5.3.23.

Answer.
56

3.5.3.24.

Answer.
The point x=c corresponds to a local maximum if A2<0 and a local minimum if A2>0.

3.5.3.25.

Answer.
1380

3.5.3.26.

Answer.
xx22+x33x44

3.5.3.27.

Answer.
xx33+x55

3.6 Taylor Series
3.6.8 Exercises

3.6.8.1.

Answer.
A: linear
B: constant
C: quadratic

3.6.8.2.

Answer.
T(5)=arctan3(e5+7)

3.6.8.3.

Answer.
A - V radius=1
B - I radius=1
C - IV radius=1
D - VI radius=+
E - II radius=+
F - III radius=+

3.6.8.4.

Answer.
(a) f(20)(3)=202(20!20!+1)
(b) g(20)(3)=102(20!10!+1)
(c) h(20)(0)=20!51111;h(22)(0)=0

3.6.8.5.

Answer.
n=1(1)n1n(x1)n

3.6.8.6.

Answer.
n=0(1)n+1(2n+1)!(xπ)2n+1

3.6.8.7.

Answer.
110n=0(10x10)n with interval of convergence (0,20).

3.6.8.8.

Answer.
n=03ne3an!(xa)n, with infinite radius of convergence

3.6.8.9. (✳).

Answer.
n=02nxn

3.6.8.10. (✳).

Answer.
bn=3(1)n+2n

3.6.8.11. (✳).

Answer.
c5=355!

3.6.8.12. (✳).

Answer.
n=0(1)n2n+1xn+1n+1 for all |x|<12

3.6.8.13. (✳).

Answer.
a=1, b=13!=16.

3.6.8.14. (✳).

Answer.
ex21xdx=Cx22+x48+.
It is not clear from the wording of the question whether or not the arbitrary constant C is to be counted as one of the “first two nonzero terms”.

3.6.8.15. (✳).

Answer.
n=0(1)n22n+1x2n+6(2n+1)(2n+6)+C=n=0(1)n22nx2n+6(2n+1)(n+3)+C

3.6.8.16. (✳).

Answer.
f(x)=1+n=0(1)n3n3n+2x3n+2

3.6.8.17. (✳).

Answer.
π23

3.6.8.18. (✳).

Answer.
1e

3.6.8.19. (✳).

Answer.
e1/e

3.6.8.20. (✳).

Answer.
e1/π1

3.6.8.21. (✳).

Answer.
log(3/2)

3.6.8.22. (✳).

Answer.
(e+2)ee2

3.6.8.23.

Answer.
The sum diverges — see the solution.

3.6.8.24.

Answer.
1+22

3.6.8.25. (✳).

Answer.
(a) See the solution.
(b) 12(e+1e)

3.6.8.26.

Answer.
(a) 50,000
(b) three terms (n=0 to n=2)
(c) six terms (n=0 to n=5)

3.6.8.27.

Answer.
29

3.6.8.28.

Answer.
S13 or higher

3.6.8.29.

Answer.
S9 or higher

3.6.8.30.

Answer.
S18 or higher

3.6.8.31.

Answer.
The error is in the interval (571437[1+137],57767)(0.199,0.040)

3.6.8.32. (✳).

Answer.
1

3.6.8.33. (✳).

Answer.
15!=1120

3.6.8.34.

Answer.
e2

3.6.8.35.

Answer.
e

3.6.8.36.

Answer.
2(6/7)3=343108

3.6.8.37.

Answer.
n=0(1)nx2n+4(2n+1)(2n+2)=x3arctanxx22log(1+x2)

3.6.8.38.

Answer.
(a) the Maclaurin series for f(x) is n=0(2n)!22n(n!)2xn, and its radius of convergence is R=1.
(b) the Maclaurin series for arcsinx is n=0(2n)!22n(n!)2(2n+1)x2n+1, and its radius of convergence is R=1.

3.6.8.39. (✳).

Answer.
log(x)=log2+n=1(1)n1n2n(x2)n. It converges when 0<x4.

3.6.8.40. (✳).

Answer.
(a) n=0(1)n x4n+14n+1
(b) 0.493967
(c) The approximate value of part (b) is larger than the true value of I(1/2)

3.6.8.41. (✳).

Answer.
166

3.6.8.42. (✳).

Answer.
Any interval of length 0.0002 that contains 0.03592 and 0.03600 is fine.

3.6.8.43. (✳).

Answer.
(a) n=1(1)n xnn n!
(b)0.80
(c) See the solution.

3.6.8.44. (✳).

Answer.
(a) Σ(x)=n=0(1)n x2n+1(2n+1)(2n+1)!
(b) x=π
(c) 1.8525

3.6.8.45. (✳).

Answer.
(a) I(x)=n=1(1)nx2n1(2n)!(2n1)
(b) I(1)=12+14!3±16!5=0.486±0.001
(c) I(1)<12+14!3

3.6.8.46. (✳).

Answer.
(a) I(x)=n=1(1)n+1x2n1(2n)!=12!x14!x3+16!x518!x8+
(b) 0.460
(c) I(1)<12!14!+16!<0.460

3.6.8.47. (✳).

Answer.
(a) See the solution.
(b) The series converges for all x.

3.6.8.48. (✳).

Answer.
See the solution.

3.6.8.49. (✳).

Answer.
(a) cosh(x)=n=0n evenxnn!=n=0x2n(2n)! for all x.
(b), (c) See the solution.

3.6.8.50.

Answer.
(a) 331.26
(b) 12 terms (S11)

3.6.8.51.

Answer.
15!5!5621!7!11!511+27!9!17!51733!11!23!523

3.6.8.52.

Answer.
(a)
(b) the constant function 0
(c) everywhere
(d) only at x=0

3.6.8.53.

Answer.
0