626
||||
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
SOLUTION We use a graphing device to produce the graphs for the cases a 苷 ⫺2, ⫺1,
⫺0.5, ⫺0.2, 0, 0.5, 1, and 2 shown in Figure 17. Notice that all of these curves (except
the case a 苷 0) have two branches, and both branches approach the vertical asymptote
x 苷 a as x approaches a from the left or right.
a=_2
a=_1
a=0
a=0.5
FIGURE 17 Members of the family
x=a+cos t, y=a tan t+sin t,
all graphed in the viewing rectangle
关_4, 4兴 by 关_4, 4兴
10.1
a=_0.5
a=_0.2
a=1
a=2
When a ⬍ ⫺1, both branches are smooth; but when a reaches ⫺1, the right branch
acquires a sharp point, called a cusp. For a between ⫺1 and 0 the cusp turns into a loop,
which becomes larger as a approaches 0. When a 苷 0, both branches come together and
form a circle (see Example 2). For a between 0 and 1, the left branch has a loop, which
shrinks to become a cusp when a 苷 1. For a ⬎ 1, the branches become smooth again,
and as a increases further, they become less curved. Notice that the curves with a positive are reflections about the y-axis of the corresponding curves with a negative.
These curves are called conchoids of Nicomedes after the ancient Greek scholar
Nicomedes. He called them conchoids because the shape of their outer branches
M
resembles that of a conch shell or mussel shell.
EXERCISES
1– 4 Sketch the curve by using the parametric equations to plot
8. x 苷 1 ⫹ 3t,
points. Indicate with an arrow the direction in which the curve is
traced as t increases.
9. x 苷 st ,
1. x 苷 1 ⫹ st ,
y 苷 t 2 ⫺ 4 t,
2. x 苷 2 cos t,
y 苷 t ⫺ cos t,
3. x 苷 5 sin t,
y 苷 t , ⫺␲ 艋 t 艋 ␲
⫺t
4. x 苷 e
⫹ t,
0艋t艋5
0 艋 t 艋 2␲
2
y 苷 e ⫺ t,
t
⫺2 艋 t 艋 2
5–10
(a) Sketch the curve by using the parametric equations to plot
points. Indicate with an arrow the direction in which the curve
is traced as t increases.
(b) Eliminate the parameter to find a Cartesian equation of
the curve.
5. x 苷 3t ⫺ 5 ,
6. x 苷 1 ⫹ t,
7. x 苷 t 2 ⫺ 2,
y 苷 2t ⫹ 1
y 苷 5 ⫺ 2t, ⫺2 艋 t 艋 3
y 苷 5 ⫺ 2t, ⫺3 艋 t 艋 4
10. x 苷 t ,
2
y 苷 2 ⫺ t2
y苷1⫺t
y 苷 t3
11–18
(a) Eliminate the parameter to find a Cartesian equation of the
curve.
(b) Sketch the curve and indicate with an arrow the direction in
which the curve is traced as the parameter increases.
11. x 苷 sin ␪,
y 苷 cos ␪, 0 艋 ␪ 艋 ␲
12. x 苷 4 cos ␪,
13. x 苷 sin t,
y 苷 5 sin ␪, ⫺␲兾2 艋 ␪ 艋 ␲兾2
y 苷 csc t,
14. x 苷 e ⫺ 1,
t
y苷e
0 ⬍ t ⬍ ␲兾2
2t
15. x 苷 e ,
y苷t⫹1
16. x 苷 ln t,
y 苷 st , t 艌 1
2t
17. x 苷 sinh t,
y 苷 cosh t
SECTION 10.1 CURVES DEFINED BY PARAMETRIC EQUATIONS
18. x 苷 2 cosh t,
y 苷 5 sinh t
||||
627
25–27 Use the graphs of x 苷 f 共t兲 and y 苷 t共t兲 to sketch the
parametric curve x 苷 f 共t兲, y 苷 t共t兲. Indicate with arrows the
direction in which the curve is traced as t increases.
19–22 Describe the motion of a particle with position 共x, y兲 as
25.
x
y
t varies in the given interval.
19. x 苷 3 ⫹ 2 cos t,
20. x 苷 2 sin t,
y 苷 4 ⫹ cos t,
21. x 苷 5 sin t,
y 苷 2 cos t,
22. x 苷 sin t,
y 苷 cos2 t,
1
␲兾2 艋 t 艋 3␲兾2
y 苷 1 ⫹ 2 sin t,
1
0 艋 t 艋 3␲兾2
t
t
1
t
_1
⫺␲ 艋 t 艋 5␲
26.
⫺2␲ 艋 t 艋 2␲
x
y
1
1
1
23. Suppose a curve is given by the parametric equations x 苷 f 共t兲,
y 苷 t共t兲, where the range of f is 关1, 4兴 and the range of t is
关2 , 3兴. What can you say about the curve?
27.
y
1
y 苷 t共t兲 in (a)–(d) with the parametric curves labeled I–IV.
Give reasons for your choices.
(a)
t
x
1
24. Match the graphs of the parametric equations x 苷 f 共t兲 and
1
1 t
t
I
y
y
1
x
2
1
2
28. Match the parametric equations with the graphs labeled I-VI.
1
1
1
t
Give reasons for your choices. (Do not use a graphing
device.)
(a) x 苷 t 4 ⫺ t ⫹ 1, y 苷 t 2
(b) x 苷 t 2 ⫺ 2t, y 苷 st
(c) x 苷 sin 2t, y 苷 sin共t ⫹ sin 2t兲
(d) x 苷 cos 5t, y 苷 sin 2t
(e) x 苷 t ⫹ sin 4t, y 苷 t 2 ⫹ cos 3t
sin 2t
cos 2t
(f) x 苷
, y苷
4 ⫹ t2
4 ⫹ t2
2 x
t
(b)
II
x
2
y
2
y
2
1t
1t
2 x
I
II
(c)
III
y
y
y
III
x
2
y
x
y
1
2
x
x
2 t
1
2 t
2 x
IV
V
y
(d)
VI
y
y
IV
x
2
y
y
2
x
2
x
2 t
x
3
5
; 29. Graph the curve x 苷 y ⫺ 3y ⫹ y .
2 t
5
2
; 30. Graph the curves y 苷 x and x 苷 y共 y ⫺ 1兲 and find their
2 x
points of intersection correct to one decimal place.
628
||||
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
31. (a) Show that the parametric equations
x 苷 x 1 ⫹ 共x 2 ⫺ x 1 兲t
41. If a and b are fixed numbers, find parametric equations for
y 苷 y1 ⫹ 共 y 2 ⫺ y1 兲t
where 0 艋 t 艋 1, describe the line segment that joins the
points P1共x 1, y1 兲 and P2共x 2 , y 2 兲.
the curve that consists of all possible positions of the point P
in the figure, using the angle ␪ as the parameter. Then eliminate the parameter and identify the curve.
y
(b) Find parametric equations to represent the line segment
from 共⫺2, 7兲 to 共3, ⫺1兲.
; 32. Use a graphing device and the result of Exercise 31(a) to
a
draw the triangle with vertices A 共1, 1兲, B 共4, 2兲, and C 共1, 5兲.
b
P
¨
33. Find parametric equations for the path of a particle that
x
O
moves along the circle x 2 ⫹ 共 y ⫺ 1兲2 苷 4 in the manner
described.
(a) Once around clockwise, starting at 共2, 1兲
(b) Three times around counterclockwise, starting at 共2, 1兲
(c) Halfway around counterclockwise, starting at 共0, 3兲
42. If a and b are fixed numbers, find parametric equations for
; 34. (a) Find parametric equations for the ellipse
x 2兾a 2 ⫹ y 2兾b 2 苷 1. [Hint: Modify the equations of
the circle in Example 2.]
(b) Use these parametric equations to graph the ellipse when
a 苷 3 and b 苷 1, 2, 4, and 8.
(c) How does the shape of the ellipse change as b varies?
the curve that consists of all possible positions of the point P
in the figure, using the angle ␪ as the parameter. The line segment AB is tangent to the larger circle.
y
A
; 35–36 Use a graphing calculator or computer to reproduce the
picture.
35.
a
y
36.
P
b
y
¨
O
x
4
2
2
0
B
2
x
0
3
8
x
43. A curve, called a witch of Maria Agnesi, consists of all pos37–38 Compare the curves represented by the parametric equa-
tions. How do they differ?
37. (a) x 苷 t 3,
y 苷 t2
(c) x 苷 e⫺3t, y 苷 e⫺2t
(b) x 苷 t 6,
y 苷 t ⫺2
t
(c) x 苷 e , y 苷 e⫺2t
(b) x 苷 cos t,
38. (a) x 苷 t,
y 苷 t4
y 苷 sec2 t
sible positions of the point P in the figure. Show that parametric equations for this curve can be written as
x 苷 2a cot ␪
Sketch the curve.
y 苷 2a sin 2␪
y
C
y=2a
A
39. Derive Equations 1 for the case ␲兾2 ⬍ ␪ ⬍ ␲.
P
a
40. Let P be a point at a distance d from the center of a circle of
radius r. The curve traced out by P as the circle rolls along a
straight line is called a trochoid. (Think of the motion of a
point on a spoke of a bicycle wheel.) The cycloid is the special case of a trochoid with d 苷 r. Using the same parameter
␪ as for the cycloid and, assuming the line is the x-axis and
␪ 苷 0 when P is at one of its lowest points, show that
parametric equations of the trochoid are
x 苷 r ␪ ⫺ d sin ␪
y 苷 r ⫺ d cos ␪
Sketch the trochoid for the cases d ⬍ r and d ⬎ r.
¨
x
O
44. (a) Find parametric equations for the set of all points P as
ⱍ
ⱍ ⱍ ⱍ
shown in the figure such that OP 苷 AB . (This curve
is called the cissoid of Diocles after the Greek scholar
Diocles, who introduced the cissoid as a graphical method
for constructing the edge of a cube whose volume is twice
that of a given cube.)
636
||||
CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
about the x-axis. Therefore, from Formula 7, we get
S 苷 y 2 r sin t s共r sin t兲2 共r cos t兲2 dt
0
0
0
苷 2 y r sin t sr 2共sin 2 t cos 2 t兲 dt 苷 2 y r sin t ⴢ r dt
]
苷 2r 2 y sin t dt 苷 2r 2共cos t兲 0 苷 4 r 2
0
10.2
M
EXERCISES
1–2 Find dy兾dx.
1. x 苷 t sin t,
y苷t t
2. x 苷 1兾t,
2
y 苷 st e
t
3–6 Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.
3. x 苷 t 4 1,
y 苷 t 3 t ; t 苷 1
4. x 苷 t t 1,
y 苷 1 t 2; t 苷 1
5. x 苷 e st ,
y 苷 sin 2
20. x 苷 cos 3,
y 苷 2 sin ; 21. Use a graph to estimate the coordinates of the rightmost point
on the curve x 苷 t t 6, y 苷 e t. Then use calculus to find the
exact coordinates.
; 22. Use a graph to estimate the coordinates of the lowest point and
the leftmost point on the curve x 苷 t 4 2t, y 苷 t t 4. Then
find the exact coordinates.
y 苷 t ln t 2 ; t 苷 1
6. x 苷 cos sin 2,
19. x 苷 2 cos ,
y 苷 sin cos 2 ; 苷 0
; 23–24 Graph the curve in a viewing rectangle that displays all the
7– 8 Find an equation of the tangent to the curve at the given point
by two methods: (a) without eliminating the parameter and (b) by
first eliminating the parameter.
7. x 苷 1 ln t,
y 苷 t 2 2; 共1, 3兲
(1, s2 )
y 苷 sec ;
8. x 苷 tan ,
important aspects of the curve.
23. x 苷 t 4 2t 3 2t 2,
y 苷 t3 t
24. x 苷 t 4 4t 3 8t 2,
y 苷 2t 2 t
25. Show that the curve x 苷 cos t, y 苷 sin t cos t has two tangents
at 共0, 0兲 and find their equations. Sketch the curve.
; 9–10 Find an equation of the tangent(s) to the curve at the given
point. Then graph the curve and the tangent(s).
y 苷 t2 t;
9. x 苷 6 sin t,
10. x 苷 cos t cos 2t,
discover where it crosses itself. Then find equations of both
tangents at that point.
共0, 0兲
y 苷 sin t sin 2t ; 共1, 1兲
2
27. (a) Find the slope of the tangent line to the trochoid
2
11–16 Find dy兾dx and d y兾dx . For which values of t is the curve
concave upward?
11. x 苷 4 t 2,
y 苷 t2 t3
13. x 苷 t e ,
y苷te
t
15. x 苷 2 sin t,
y 苷 3 cos t,
16. x 苷 cos 2t ,
y 苷 cos t ,
t
; 26. Graph the curve x 苷 cos t 2 cos 2t, y 苷 sin t 2 sin 2t to
12. x 苷 t 3 12t,
y 苷 t2 1
14. x 苷 t ln t,
y 苷 t ln t
0 t 2
0t
x 苷 r d sin , y 苷 r d cos in terms of . (See Exercise 40 in Section 10.1.)
(b) Show that if d r, then the trochoid does not have a
vertical tangent.
28. (a) Find the slope of the tangent to the astroid x 苷 a cos 3,
y 苷 a sin 3 in terms of . (Astroids are explored in the
Laboratory Project on page 629.)
(b) At what points is the tangent horizontal or vertical?
(c) At what points does the tangent have slope 1 or 1?
29. At what points on the curve x 苷 2t 3, y 苷 1 4t t 2 does the
17–20 Find the points on the curve where the tangent is horizontal
or vertical. If you have a graphing device, graph the curve to check
your work.
17. x 苷 10 t ,
2
y 苷 t 12t
3
18. x 苷 2t 3 3t 2 12t,
y 苷 2t 3 3t 2 1
tangent line have slope 1?
30. Find equations of the tangents to the curve x 苷 3t 2 1,
y 苷 2t 3 1 that pass through the point 共4, 3兲.
31. Use the parametric equations of an ellipse, x 苷 a cos ,
y 苷 b sin , 0 2, to find the area that it encloses.
SECTION 10.2 CALCULUS WITH PARAMETRIC CURVES
637
49. Use Simpson’s Rule with n 苷 6 to estimate the length of the
32. Find the area enclosed by the curve x 苷 t 2 2t, y 苷 st and
curve x 苷 t e t, y 苷 t e t, 6 t 6.
the y-axis.
50. In Exercise 43 in Section 10.1 you were asked to derive the
33. Find the area enclosed by the x-axis and the curve
parametric equations x 苷 2a cot , y 苷 2a sin 2 for the
curve called the witch of Maria Agnesi. Use Simpson’s Rule
with n 苷 4 to estimate the length of the arc of this curve
given by 兾4 兾2.
x 苷 1 e t, y 苷 t t 2.
34. Find the area of the region enclosed by the astroid
x 苷 a cos 3, y 苷 a sin 3. (Astroids are explored in the Laboratory Project on page 629.)
_a
||||
y
51–52 Find the distance traveled by a particle with position 共x, y兲
a
as t varies in the given time interval. Compare with the length of
the curve.
a
0
x
0 t 3
51. x 苷 sin 2 t,
y 苷 cos 2 t,
52. x 苷 cos 2t,
y 苷 cos t, 0 t 4
53. Show that the total length of the ellipse x 苷 a sin ,
_a
y 苷 b cos , a b 0, is
35. Find the area under one arch of the trochoid of Exercise 40 in
L 苷 4a y
Section 10.1 for the case d r.
36. Let ᏾ be the region enclosed by the loop of the curve in
Example 1.
(a) Find the area of ᏾.
(b) If ᏾ is rotated about the x-axis, find the volume of the
resulting solid.
(c) Find the centroid of ᏾.
37. x 苷 t t 2,
y 苷 43 t 3兾2, 1 t 2
38. x 苷 1 e t,
y 苷 t 2,
54. Find the total length of the astroid x 苷 a cos 3, y 苷 a sin 3,
where a 0.
CAS
55. (a) Graph the epitrochoid with equations
x 苷 11 cos t 4 cos共11t兾2兲
40. x 苷 ln t,
y 苷 11 sin t 4 sin共11t兾2兲
What parameter interval gives the complete curve?
(b) Use your CAS to find the approximate length of this
curve.
3 t 3
y 苷 t sin t,
s1 e 2 sin 2 d
where e is the eccentricity of the ellipse (e 苷 c兾a, where
c 苷 sa 2 b 2 ) .
37– 40 Set up an integral that represents the length of the curve.
Then use your calculator to find the length correct to four decimal
places.
39. x 苷 t cos t,
兾2
0
0 t 2
CAS
56. A curve called Cornu’s spiral is defined by the parametric
equations
y 苷 st 1, 1 t 5
t
x 苷 C共t兲 苷 y cos共 u 2兾2兲 du
0
41– 44 Find the exact length of the curve.
t
41. x 苷 1 3t 2,
y 苷 4 2t 3,
42. x 苷 e t et,
y 苷 5 2t,
0t3
y 苷 ln共1 t兲,
0t2
t
,
43. x 苷
1t
44. x 苷 3 cos t cos 3t,
y 苷 S共t兲 苷 y sin共 u 2兾2兲 du
0t1
y 苷 3 sin t sin 3t,
0
0t
; 45– 47 Graph the curve and find its length.
45. x 苷 e t cos t,
y 苷 e t sin t, 0 t 46. x 苷 cos t ln(tan 2 t),
1
47. x 苷 e t,
t
y 苷 4e ,
t兾2
y 苷 sin t, 兾4 t 3兾4
8 t 3
48. Find the length of the loop of the curve x 苷 3t t 3,
y 苷 3t 2.
where C and S are the Fresnel functions that were introduced
in Chapter 5.
(a) Graph this curve. What happens as t l and as
t l ?
(b) Find the length of Cornu’s spiral from the origin to the
point with parameter value t.
57–58 Set up an integral that represents the area of the surface
obtained by rotating the given curve about the x-axis. Then use
your calculator to find the surface area correct to four decimal
places.
57. x 苷 1 te t,
58. x 苷 sin 2 t,
y 苷 共t 2 1兲e t,
0t1
y 苷 sin 3t, 0 t 兾3
APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES
5.
Species 2
15. (a) P共t兲 苷
t=2
200
t=3
23. (a) Stabilizes at 200,000
t=4
50
t=0, 5
0
50
100
150
200
250 Species 1
9. (a) Population stabilizes at 5000.
(b) (i) W 苷 0, R 苷 0: Zero populations
(ii) W 苷 0, R 苷 5000: In the absence of wolves, the rabbit
population is always 5000.
(iii) W 苷 64, R 苷 1000: Both populations are stable.
(c) The populations stabilize at 1000 rabbits and 64 wolves.
R
(b) (i) x 苷 0, y 苷 0: Zero populations
(ii) x 苷 200,000, y 苷 0: In the absence of birds, the insect
population is always 200,000.
(iii) x 苷 25,000, y 苷 175: Both populations are stable.
(c) The populations stabilize at 25,000 insects and 175 birds.
(d)
x (insects)
(birds) y
45,000
insects
100
15,000
50
0
t
25. (a) y 苷 共1兾k兲 cosh kx a 1兾k or
y 苷 共1兾k兲 cosh kx 共1兾k兲 cosh kb h
40
R
500
200
150
5,000
60
1000
birds
25,000
80
W
250
35,000
W
1500
(b) L共t兲 苷 53 43e0.2t
21. k ln h h 苷 共R兾V 兲t C
19. 15 days
t=1
100
(b) 共2兾k兲 sinh kb
20
0
PROBLEMS PLUS
t
1. f 共x兲 苷 10e
CHAPTER 9 REVIEW
N
PAGE 615
9. (b) f 共x兲 苷
5. True
7. True
冉冊
2
7. 20 C
(c) No
(b) 31,900 ⬇ 100,000 ft 2; 6283 ft 2兾h
(c) 5.1 h
13. x 2 共 y 6兲2 苷 25
Exercises
1. (a)
5. y 苷 x 1兾n
x L
x
12 L ln
4L
L
11. (a) 9.8 h
3. False
PAGE 618
x
2
True-False Quiz
1. True
N
(b) 0 c 4;
y 苷 0, y 苷 2, y 苷 4
y
6
CHAPTER 10
(iv) 4
(iii)
EXERCISES 10.1
2
1.
N
PAGE 626
3.
y
t=5
5, 5}
{1+œ„
(ii)
(i) 0
1
3. (a)
t=π {0, π@}
5
t=4
(3, 0)
x
y共0.3兲 ⬇ 0.8
y
3
y
t
t=0
(1, 0)
t=0 (0, 0)
2
1
5. (a)
_3
_2
_1
0
1
2
3x
(b) 0.75676
(c) y 苷 x and y 苷 x; there is a local maximum or minimum
1
5. y 苷 (2 x 2 C) esin x
7. y 苷 sln共x 2 2x 3兾2 C兲
2
1
1
9. r共t兲 苷 5e tt
11. y 苷 2 x 共ln x兲2 2x
13. x 苷 C 2 y 2
(b) y 苷 23 x 133
y
(1, 5)
t=2
(_2, 3)
t=1
(_5, 1)
t=0
(_8, _1)
t=_1
0
x
A101
(b) t 苷 10 ln 572 ⬇ 33.5
17. (a) L共t兲 苷 L 关L L共0兲兴ekt
150
(d)
2000
; ⬇560
1 19e0.1t
||||
5 x
A102
||||
APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES
7. (a)
(b) x 苷 4 共 y 5兲2 2,
3 y 11
1
y
(7 , 11)
t=_3
(_2, 5)
t=0
x
1
”4 4 , 0’
9. (a)
5
t=2
(14, _3)
t=4
(0, 1) t=0
(b) x 苷 2 cos t, y 苷 1 2 sin t, 0 t 6
(c) x 苷 2 cos t, y 苷 1 2 sin t, 兾2 t 3兾2
37. The curve y 苷 x 2兾3 is generated in (a). In (b), only the portion
with x 0 is generated, and in (c) we get only the portion with
x 0.
41. x 苷 a cos , y 苷 b sin ; 共x 2兾a 2 兲 共 y 2兾b 2 兲 苷 1, ellipse
43.
(b) y 苷 1 x 2, x 0
y
31. (b) x 苷 2 5t, y 苷 7 8t, 0 t 1
33. (a) x 苷 2 cos t, y 苷 1 2 sin t, 0 t 2
y
2a
(1, 0) t=1
x
O
x
0
45. (a) Two points of intersection
4
(2, _3) t=4
11. (a) x 2 y 2 苷 1, x 0
13. (a) y 苷 1兾x, y 1
(b)
(b)
y
(0, 1)
6
6
y
4
(b) One collision point at 共3, 0兲 when t 苷 3兾2
(c) There are still two intersection points, but no collision point.
47. For c 苷 0, there is a cusp; for c 0, there is a loop whose size
increases as c increases.
(1, 1)
x
0
0
x
(0, _1)
3
15. (a) y 苷 2 ln x 1
(b)
1
17. (a) y 2 x 2 苷 1, y 1
1
(b)
y
_1
1
2
1
y
0
0
1.5
0
1.5
1
0
1
x
1
_3
x
0
19. Moves counterclockwise along the circle
共x 3兲2 共 y 1兲2 苷 4 from 共3, 3兲 to 共3, 1兲
21. Moves 3 times clockwise around the ellipse
共x 2兾25兲 共 y 2兾4兲 苷 1, starting and ending at 共0, 2兲
23. It is contained in the rectangle described by 1 x 4
and 2 y 3.
25.
27.
y
y
1
1
t= 2
_1
49. As n increases, the number of oscillations increases;
a and b determine the width and height.
EXERCISES 10.2
N
PAGE 636
2t 1
3. y 苷 x
t cos t sin t
5. y 苷 共2兾e兲x 3
7. y 苷 2x 1
1
9. y 苷 6 x
20
1.
(0, 1) t=1
(_1, 0)
t=0
x
t=0
1
x
(0, _1) t=_1
_10
10
_2
29.
3
11. 1 2 t, 3兾共4t兲, t 0
13. et, et兾共1 e t 兲, t 0
3
3
15. 2 tan t, 4 sec 3 t, 兾2 t 3兾2
3
3
3
17. Horizontal at 共6, 16兲, vertical at 共10, 0兲
19. Horizontal at (s2 , 1) (four points), vertical at 共2, 0兲
3
21. 共0.6, 2兲; (5 ⴢ 6 6兾5, e 6
1兾5
)
APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES
23.
25. y 苷 x, y 苷 x
7.5
||||
A103
(c)
y
π
2
O
π
8.5
3
”_1, 2 ’
x
0
1
共1, 3 兾2兲, 共1, 5兾2兲
27. (a) d sin 兾共r d cos 兲
29. ( 27, 9 ), 共2, 4兲
31. ab
33. 3 e
35. 2r 2 d 2
16 29
37.
2
1
x
3. (a)
π
(1, π)
s1 4t dt ⬇ 3.1678
O
2
O
_ 2π
x02 s3 2 sin t 2 cos t
41. 4s2 2
dt ⬇ 10.0367
43. s10兾3 ln(3 s10 ) s2 ln(1 s2 )
45. s2 共e 1兲
8
39.
(b)
3
”2, _ 2π ’
3
(1, s3 )
共1, 0兲
(c)
3π
4
O
25
0
47. e 3 11 e8
2.5
3π
21
”_2, 4 ’
(s2, s2 )
5. (a) (i) (2s2, 7兾4)
(b) (i) 共2, 2兾3兲
(ii) (2s2, 3兾4)
(ii) 共2, 5兾3兲
7.
9.
π
¨= 6
r=2
1
49. 612.3053
55. (a)
21
r=1
1
O
O
51. 6 s2, s2
π
¨=_ 2
t 僆 关0, 4兴
15
11.
15
¨=
15
r=4
7π
3
r=3
r=2
O
15
(b) ⬇ 294
57.
59.
65.
x01 2 共t 2 1兲e tse 2t 共t 1兲 2共t 2 2t 2兲 dt ⬇ 103.5999
2
1215
24
5
(247 s13 64)
(949 s26 1)
EXERCISES 10.3
N
61. 5 a 2
6
71.
63. 59.101
1
4
PAGE 647
1. (a)
π
”2, 3 ’
(b)
13.
17.
19.
21.
27.
29.
15. Circle, center O, radius 2
2s3
3
3
Circle, center (0, 2 ), radius 2
Horizontal line, 1 unit above the x-axis
r 苷 3 sec 23. r 苷 cot csc (a) 苷 兾6
(b) x 苷 3
25. r 苷 2c cos 31.
π
O
”1, _ 3π ’
4
_ 3π
4
O
π
¨=_ 6
共2, 7兾3兲, 共2, 4兾3兲
5π
3
”1, 2 ’
π
3
O
¨=
共1, 5兾4兲, 共1, 兾4兲
O