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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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Chapter
Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to determine the parameter interval.

53. r = sin(6 sinθ)

To determine

To Find: The length of the curve.

Explanation

Given:

The polar equation is r=sin(6sinθ) (1)

Formula used:

The exact length of the polar curve is,

L=abr2+(drdθ)2dθ (2)

Calculation:

It is given that the polar equation is r=sin(6sinθ).

Substitute 0 for θ in r=sin(6sinθ),

r=sin(6sin(0))=0

The polar equations are x=rcosθ and y=rsinθ.

Substitute 0 for r and 0 for θ in x=rcosθ,

x=rcosθ=0×cos(0×π180)=0

Substitute 0 for r and 0 for θ y=rsinθ,

y=0×sin(0×π180)=0

Repeat the calculation of the values of x and y using the value of θ from 0 to π.

Tabulate the values of x and y in table (1).

θr=sin(6sinθ)x=rcosθy=rsinθ
0.000.000.000.00
1.000.100.100.00
2.000.210.210.01
3.000.310.310.02
4.000.410.410.03
5.000.500.500.04
6.000.590.580.06
7.000.670.660.08
8.000.740.730.10
9.000.810.800.13
10.000.860.850.15
11.000.910.890.17
12.000.950.930.20
13.000.980.950.22
14.000.990.960.24
15.001.000.970.26
16.001.000.960.27
17.000.980.940.29
18.000.960.910.30
19.000.930.880.30
20.000.890.830.30
21.000.840.780.30
22.000.780.720.29
23.000.720.660.28
24.000.650.590.26
25.000.570.520.24
26.000.490.440.21
27.000.410.360.18
28.000.320.280.15
29.000.230.200.11
30.000.140.120.07
31.000.050.040.03
32.00-0.04-0.03-0.02
33.00-0.13-0.11-0.07
34.00-0.21-0.18-0.12
35.00-0.30-0.24-0.17
36.00-0.38-0.30-0.22
37.00-0.45-0.36-0.27
38.00-0.52-0.41-0.32
39.00-0.59-0.46-0.37
40.00-0.66-0.50-0.42
41.00-0.71-0.54-0.47
42.00-0.77-0.57-0.51
43.00-0.81-0.60-0.55
44.00-0.86-0.62-0.59
45.00-0.89-0.63-0.63
46.00-0.92-0.64-0.66
47.00-0.95-0.65-0.69
48.00-0.97-0.65-0.72
49.00-0.98-0.64-0.74
50.00-0.99-0.64-0.76
51.00-1.00-0.63-0.78
52.00-1.00-0.62-0.79
53.00-1.00-0.60-0.80
54.00-0.99-0.58-0.80
55.00-0.98-0.56-0.80
56.00-0.97-0.54-0.80
57.00-0.95-0.52-0.80
58.00-0.93-0.49-0.79
59.00-0.91-0.47-0.78
60.00-0.89-0.44-0.77
61.00-0.86-0.42-0.75
62.00-0.83-0.39-0.74
63.00-0.81-0.37-0.72
64.00-0.78-0.34-0.70
65.00-0.75-0.32-0.68
66.00-0.72-0.29-0.66
67.00-0.69-0.27-0.63
68.00-0.66-0.25-0.61
69.00-0.63-0.23-0.59
70.00-0.60-0.21-0.56
71.00-0.57-0.19-0.54
72.00-0.55-0.17-0.52
73.00-0.52-0.15-0.50
74.00-0.49-0.14-0.47
75.00-0.47-0.12-0.45
76.00-0.45-0.11-0.43
77.00-0.42-0.10-0.41
78.00-0.40-0.08-0.39
79.00-0.38-0.07-0.38
80.00-0.37-0.06-0.36
81.00-0.35-0.05-0.35
82.00-0.33-0.05-0.33
83.00-0.32-0.04-0.32
84.00-0.31-0.03-0.31
85.00-0.30-0.03-0.30
86.00-0.29-0.02-0.29
87.00-0.29-0.02-0.29
88.00-0.28-0.01-0.28
89.00-0.280.00-0.28
90.00-0.280.00-0.28
91.00-0.280.00-0.28
92.00-0.280.01-0.28
93.00-0.290.02-0.29
94.00-0.290.02-0.29
95.00-0.300.03-0.30
96.00-0.310.03-0.31
97.00-0.320.04-0.32
98.00-0.330.05-0.33
99.00-0.350.05-0.35
100.00-0.370.06-0.36
101.00-0.380.07-0

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P-19ESect-10.3 P-20ESect-10.3 P-21ESect-10.3 P-22ESect-10.3 P-23ESect-10.3 P-24ESect-10.3 P-25ESect-10.3 P-26ESect-10.3 P-27ESect-10.3 P-28ESect-10.3 P-29ESect-10.3 P-30ESect-10.3 P-31ESect-10.3 P-32ESect-10.3 P-33ESect-10.3 P-34ESect-10.3 P-35ESect-10.3 P-36ESect-10.3 P-37ESect-10.3 P-38ESect-10.3 P-39ESect-10.3 P-40ESect-10.3 P-41ESect-10.3 P-42ESect-10.3 P-43ESect-10.3 P-44ESect-10.3 P-45ESect-10.3 P-46ESect-10.3 P-47ESect-10.3 P-48ESect-10.3 P-49ESect-10.3 P-50ESect-10.3 P-51ESect-10.3 P-52ESect-10.3 P-53ESect-10.3 P-54ESect-10.3 P-55ESect-10.3 P-56ESect-10.3 P-57ESect-10.3 P-58ESect-10.3 P-59ESect-10.3 P-60ESect-10.3 P-61ESect-10.3 P-62ESect-10.3 P-63ESect-10.3 P-64ESect-10.3 P-65ESect-10.3 P-66ESect-10.3 P-67ESect-10.3 P-68ESect-10.3 P-69ESect-10.3 P-70ESect-10.3 P-71ESect-10.3 P-72ESect-10.3 P-73ESect-10.3 P-74ESect-10.3 P-75ESect-10.3 P-76ESect-10.3 P-77ESect-10.3 P-78ESect-10.4 P-1ESect-10.4 P-2ESect-10.4 P-3ESect-10.4 P-4ESect-10.4 P-5ESect-10.4 P-6ESect-10.4 P-7ESect-10.4 P-8ESect-10.4 P-9ESect-10.4 P-10ESect-10.4 P-11ESect-10.4 P-12ESect-10.4 P-13ESect-10.4 P-14ESect-10.4 P-15ESect-10.4 P-16ESect-10.4 P-17ESect-10.4 P-18ESect-10.4 P-19ESect-10.4 P-20ESect-10.4 P-21ESect-10.4 P-22ESect-10.4 P-23ESect-10.4 P-24ESect-10.4 P-25ESect-10.4 P-26ESect-10.4 P-27ESect-10.4 P-28ESect-10.4 P-29ESect-10.4 P-30ESect-10.4 P-31ESect-10.4 P-32ESect-10.4 P-33ESect-10.4 P-34ESect-10.4 P-35ESect-10.4 P-36ESect-10.4 P-37ESect-10.4 P-38ESect-10.4 P-39ESect-10.4 P-40ESect-10.4 P-41ESect-10.4 P-42ESect-10.4 P-43ESect-10.4 P-44ESect-10.4 P-45ESect-10.4 P-46ESect-10.4 P-47ESect-10.4 P-48ESect-10.4 P-49ESect-10.4 P-50ESect-10.4 P-51ESect-10.4 P-52ESect-10.4 P-53ESect-10.4 P-54ESect-10.4 P-55ESect-10.4 P-56ESect-10.5 P-1ESect-10.5 P-2ESect-10.5 P-3ESect-10.5 P-4ESect-10.5 P-5ESect-10.5 P-6ESect-10.5 P-7ESect-10.5 P-8ESect-10.5 P-9ESect-10.5 P-10ESect-10.5 P-11ESect-10.5 P-12ESect-10.5 P-13ESect-10.5 P-14ESect-10.5 P-15ESect-10.5 P-16ESect-10.5 P-17ESect-10.5 P-18ESect-10.5 P-19ESect-10.5 P-20ESect-10.5 P-21ESect-10.5 P-22ESect-10.5 P-23ESect-10.5 P-24ESect-10.5 P-25ESect-10.5 P-26ESect-10.5 P-27ESect-10.5 P-28ESect-10.5 P-29ESect-10.5 P-30ESect-10.5 P-31ESect-10.5 P-32ESect-10.5 P-33ESect-10.5 P-34ESect-10.5 P-35ESect-10.5 P-36ESect-10.5 P-37ESect-10.5 P-38ESect-10.5 P-39ESect-10.5 P-40ESect-10.5 P-41ESect-10.5 P-42ESect-10.5 P-43ESect-10.5 P-44ESect-10.5 P-45ESect-10.5 P-46ESect-10.5 P-47ESect-10.5 P-48ESect-10.5 P-49ESect-10.5 P-50ESect-10.5 P-51ESect-10.5 P-52ESect-10.5 P-53ESect-10.5 P-54ESect-10.5 P-55ESect-10.5 P-56ESect-10.5 P-57ESect-10.5 P-58ESect-10.5 P-59ESect-10.5 P-60ESect-10.5 P-61ESect-10.5 P-62ESect-10.5 P-63ESect-10.5 P-64ESect-10.5 P-65ESect-10.5 P-66ESect-10.6 P-1ESect-10.6 P-2ESect-10.6 P-3ESect-10.6 P-4ESect-10.6 P-5ESect-10.6 P-6ESect-10.6 P-7ESect-10.6 P-8ESect-10.6 P-9ESect-10.6 P-10ESect-10.6 P-11ESect-10.6 P-12ESect-10.6 P-13ESect-10.6 P-14ESect-10.6 P-15ESect-10.6 P-16ESect-10.6 P-17ESect-10.6 P-18ESect-10.6 P-19ESect-10.6 P-20ESect-10.6 P-21ESect-10.6 P-22ESect-10.6 P-23ESect-10.6 P-24ESect-10.6 P-25ESect-10.6 P-26ESect-10.6 P-27ESect-10.6 P-28ESect-10.6 P-29ESect-10.6 P-30ESect-10.6 P-31ECh-10 P-1RCCCh-10 P-2RCCCh-10 P-3RCCCh-10 P-4RCCCh-10 P-5RCCCh-10 P-6RCCCh-10 P-7RCCCh-10 P-8RCCCh-10 P-9RCCCh-10 P-1RQCh-10 P-2RQCh-10 P-3RQCh-10 P-4RQCh-10 P-5RQCh-10 P-6RQCh-10 P-7RQCh-10 P-8RQCh-10 P-9RQCh-10 P-10RQCh-10 P-1RECh-10 P-2RECh-10 P-3RECh-10 P-4RECh-10 P-5RECh-10 P-6RECh-10 P-7RECh-10 P-8RECh-10 P-9RECh-10 P-10RECh-10 P-11RECh-10 P-12RECh-10 P-13RECh-10 P-14RECh-10 P-15RECh-10 P-16RECh-10 P-17RECh-10 P-18RECh-10 P-19RECh-10 P-20RECh-10 P-21RECh-10 P-22RECh-10 P-23RECh-10 P-24RECh-10 P-25RECh-10 P-26RECh-10 P-27RECh-10 P-28RECh-10 P-29RECh-10 P-30RECh-10 P-31RECh-10 P-32RECh-10 P-33RECh-10 P-34RECh-10 P-35RECh-10 P-36RECh-10 P-37RECh-10 P-38RECh-10 P-39RECh-10 P-40RECh-10 P-41RECh-10 P-42RECh-10 P-43RECh-10 P-44RECh-10 P-45RECh-10 P-46RECh-10 P-47RECh-10 P-48RECh-10 P-49RECh-10 P-50RECh-10 P-51RECh-10 P-52RECh-10 P-53RECh-10 P-54RECh-10 P-55RECh-10 P-56RECh-10 P-57RECh-10 P-58RECh-10 P-1PCh-10 P-2PCh-10 P-3PCh-10 P-4PCh-10 P-5PCh-10 P-6P

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