Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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10:29
12.3 Calculus in Polar Coordinates
70. The parabola r =
63-74. Arc length of polar curves Find the length of the following
polar curves.
63. The complete circle r = a sin 0, where a > 0
64. The complete cardioid r = 2 - 2 sin 0
65. The spiral r = 0², for 0 ≤ 0 ≤ 2π
66. The spiral reº, for 0 ≤ 0 ≤ 2πn, where n is a positive integer
67. The complete cardioid r = 4 + 4 sin 0
68. The spiral r = 40², for 0 ≤ 0 ≤6
69. The spiral = 2e20, for 0 ≤ 0 ≤ In 8.
V2
1 + cos 0'
for 0 ≤ 0 ≤
TT
71. The curve r sin³ for 0 ≤ 0 ≤
0
3'
플
T72. The three-leaf rose r = 2 cos 30
. LTE
TT
2
787
T73. The complete limaçon r = 4 - 2 cos 0
T74. The complete lemniscater² = 6 sin 20
75. Explain why or why not Determine whether the following state-
ments are true and give an explanation or counterexample.
a. The area of the region bounded by the polar graph of r = f(0)
on the interval [a, b] is f(0) do.
b. The slope of the line tangent to the polar curve r = f(0) at a
point (r, 0) is f'(0).
c. There may be more than one line that is tangent to a polar
curve at some points on the curve.
Explorations and Challenges
76. Area calculation Use polar coordinates to determine the area
bounded on the right by the unit circle x² + y² = 1 and bounded
on the left by the vertical line x = V√2/2.
77. Spiral tangent lines Use a graphing utility to determine the first
three points with ≥ 0 at which the spiral r = 20 has a horizon-
tal tangent line. Find the first three points with ≥ 0 at which the
spiral r = 20 has a vertical tangent line.
B-787
78. Spiral arc length Consider the spiral r = 40, for ≥ 0.
a. Use a trigonometric substitution to find the length of the spiral,
for 0 0
V8.
b. Find L(0), the length of the spiral on the interval [0, 0], for
any ≥ 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d314298-732f-46eb-a79a-0529aaba02f7%2F4d995e7b-cd3c-4768-8a4d-6f6199a342ba%2Fkmsoky_processed.jpeg&w=3840&q=75)
Transcribed Image Text:0
10:29
12.3 Calculus in Polar Coordinates
70. The parabola r =
63-74. Arc length of polar curves Find the length of the following
polar curves.
63. The complete circle r = a sin 0, where a > 0
64. The complete cardioid r = 2 - 2 sin 0
65. The spiral r = 0², for 0 ≤ 0 ≤ 2π
66. The spiral reº, for 0 ≤ 0 ≤ 2πn, where n is a positive integer
67. The complete cardioid r = 4 + 4 sin 0
68. The spiral r = 40², for 0 ≤ 0 ≤6
69. The spiral = 2e20, for 0 ≤ 0 ≤ In 8.
V2
1 + cos 0'
for 0 ≤ 0 ≤
TT
71. The curve r sin³ for 0 ≤ 0 ≤
0
3'
플
T72. The three-leaf rose r = 2 cos 30
. LTE
TT
2
787
T73. The complete limaçon r = 4 - 2 cos 0
T74. The complete lemniscater² = 6 sin 20
75. Explain why or why not Determine whether the following state-
ments are true and give an explanation or counterexample.
a. The area of the region bounded by the polar graph of r = f(0)
on the interval [a, b] is f(0) do.
b. The slope of the line tangent to the polar curve r = f(0) at a
point (r, 0) is f'(0).
c. There may be more than one line that is tangent to a polar
curve at some points on the curve.
Explorations and Challenges
76. Area calculation Use polar coordinates to determine the area
bounded on the right by the unit circle x² + y² = 1 and bounded
on the left by the vertical line x = V√2/2.
77. Spiral tangent lines Use a graphing utility to determine the first
three points with ≥ 0 at which the spiral r = 20 has a horizon-
tal tangent line. Find the first three points with ≥ 0 at which the
spiral r = 20 has a vertical tangent line.
B-787
78. Spiral arc length Consider the spiral r = 40, for ≥ 0.
a. Use a trigonometric substitution to find the length of the spiral,
for 0 0
V8.
b. Find L(0), the length of the spiral on the interval [0, 0], for
any ≥ 0.
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