Chapter 10.4, Problem 28E
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Single Variable Calculus: Early Tr...
8th Edition
James Stewart
ISBN: 9781305270336
Solutions
Chapter
Section
BuyFindarrow_forward
Single Variable Calculus: Early Tr...
8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem
Find the area of the region that lies inside the first curve and outside the second curve.
28. r = 3 sin θ, r = 2 − sin θ
To determine
To Find: The area of the region that lies inside the first curve and outside the second curve.
Explanation
Given:
The polar equations are as below.
Calculation:
Assume the value of
Calculate the value of r using equation (1).
Substitute
Calculate the value of x.
Substitute 0 for r and
Calculate the value of y.
Substitute 0 for r and
Similarly, calculate the values of x and y using the value of
Tabulate the values of x and y in table (1).
| 0.00 | 0.00 | 0.00 | 0.00 |
| 10.00 | 0.52 | 0.51 | 0.09 |
| 20.00 | 1.03 | 0.96 | 0.35 |
| 30.00 | 1.50 | 1.30 | 0.75 |
| 40.00 | 1.93 | 1.48 | 1.24 |
| 50.00 | 2.30 | 1.48 | 1.76 |
| 60.00 | 2.60 | 1.30 | 2.25 |
| 70.00 | 2.82 | 0.96 | 2.65 |
| 80.00 | 2.95 | 0.51 | 2.91 |
| 90.00 | 3.00 | 0.00 | 3.00 |
| 100.00 | 2.95 | -0.51 | 2.91 |
| 110.00 | 2.82 | -0.96 | 2.65 |
| 120.00 | 2.60 | -1.30 | 2.25 |
| 130.00 | 2.30 | -1.48 | 1.76 |
| 140.00 | 1.93 | -1.48 | 1.24 |
| 150.00 | 1.50 | -1.30 | 0.75 |
| 160.00 | 1.03 | -0.96 | 0.35 |
| 170.00 | 0.52 | -0.51 | 0.09 |
| 180.00 | 0.00 | 0.00 | 0.00 |
| 190.00 | -0.52 | 0.51 | 0.09 |
| 200.00 | -1.03 | 0.96 | 0.35 |
| 210.00 | -1.50 | 1.30 | 0.75 |
| 220.00 | -1.93 | 1.48 | 1.24 |
| 230.00 | -2.30 | 1.48 | 1.76 |
| 240.00 | -2.60 | 1.30 | 2.25 |
| 250.00 | -2.82 | 0.96 | 2.65 |
| 260.00 | -2.95 | 0.51 | 2.91 |
| 270.00 | -3.00 | 0.00 | 3.00 |
| 280.00 | -2.95 | -0.51 | 2.91 |
| 290.00 | -2.82 | -0.96 | 2.65 |
| 300.00 | -2.60 | -1.30 | 2.25 |
| 310.00 | -2.30 | -1.48 | 1.76 |
| 320.00 | -1.93 | -1.48 | 1.24 |
| 330.00 | -1.50 | -1.30 | 0.75 |
| 340.00 | -1.03 | -0.96 | 0.35 |
| 350.00 | -0.52 | -0.51 | 0.09 |
| 360.00 | 0.00 | 0.00 | 0.00 |
| 0.00 | 0.00 | 0.00 | 0.00 |
| 10.00 | 0.52 | 0.51 | 0.09 |
Calculate the value of r using equation (2),
Substitute
Calculate the value of x.
Substitute 2 for r and
Calculate the value of y.
Substitute 2 for r and
Similarly, calculate the values of x and y using the value of
Tabulate the values of x and y in table (2).
| 0.00 | 2.00 | 2.00 | 0.00 |
| 10.00 | 1.83 | 1.80 | 0.32 |
| 20.00 | 1.66 | 1.56 | 0.57 |
| 30.00 | 1.50 | 1.30 | 0.75 |
| 40.00 | 1.36 | 1.04 | 0.87 |
| 50.00 | 1.23 | 0.79 | 0.95 |
| 60.00 | 1.13 | 0.57 | 0.98 |
| 70.00 | 1.06 | 0.36 | 1.00 |
| 80.00 | 1.02 | 0.18 | 1.00 |
| 90.00 | 1.00 | 0.00 | 1.00 |
| 100.00 | 1.02 | -0.18 | 1.00 |
| 110.00 | 1.06 | -0.36 | 1.00 |
| 120.00 | 1.13 | -0.57 | 0.98 |
| 130.00 | 1.23 | -0.79 | 0.95 |
| 140.00 | 1.36 | -1.04 | 0.87 |
| 150.00 | 1.50 | -1.30 | 0.75 |
| 160.00 | 1.66 | -1.56 | 0.57 |
| 170.00 | 1.83 | -1.80 | 0.32 |
| 180.00 | 2.00 | -2.00 | 0.00 |
| 190.00 | 2.17 | -2.14 | -0.38 |
| 200.00 | 2.34 | -2.20 | -0.80 |
| 210.00 | 2.50 | -2.17 | -1.25 |
| 220.00 | 2.64 | -2.02 | -1.70 |
| 230.00 | 2.77 | -1.78 | -2.12 |
| 240.00 | 2.87 | -1.43 | -2.48 |
| 250.00 | 2.94 | -1.01 | -2.76 |
| 260.00 | 2.98 | -0.52 | -2.94 |
| 270.00 | 3.00 | 0.00 | -3.00 |
| 280.00 | 2.98 | 0.52 | -2.94 |
| 290.00 | 2.94 | 1.01 | -2.76 |
| 300.00 | 2.87 | 1.43 | -2.48 |
| 310.00 | 2.77 | 1.78 | -2.12 |
| 320 |
Chapter 10 Solutions
Sect-10.1 P-1ESect-10.1 P-2ESect-10.1 P-3ESect-10.1 P-4ESect-10.1 P-5ESect-10.1 P-6ESect-10.1 P-7ESect-10.1 P-8ESect-10.1 P-9ESect-10.1 P-10E
Sect-10.1 P-11ESect-10.1 P-12ESect-10.1 P-13ESect-10.1 P-14ESect-10.1 P-15ESect-10.1 P-16ESect-10.1 P-17ESect-10.1 P-18ESect-10.1 P-19ESect-10.1 P-20ESect-10.1 P-21ESect-10.1 P-22ESect-10.1 P-23ESect-10.1 P-24ESect-10.1 P-25ESect-10.1 P-26ESect-10.1 P-27ESect-10.1 P-28ESect-10.1 P-29ESect-10.1 P-30ESect-10.1 P-31ESect-10.1 P-32ESect-10.1 P-33ESect-10.1 P-34ESect-10.1 P-35ESect-10.1 P-36ESect-10.1 P-37ESect-10.1 P-38ESect-10.1 P-39ESect-10.1 P-40ESect-10.1 P-41ESect-10.1 P-42ESect-10.1 P-43ESect-10.1 P-44ESect-10.1 P-45ESect-10.1 P-46ESect-10.1 P-47ESect-10.1 P-48ESect-10.1 P-49ESect-10.1 P-50ESect-10.1 P-51ESect-10.1 P-52ESect-10.2 P-1ESect-10.2 P-2ESect-10.2 P-3ESect-10.2 P-4ESect-10.2 P-5ESect-10.2 P-6ESect-10.2 P-7ESect-10.2 P-8ESect-10.2 P-9ESect-10.2 P-10ESect-10.2 P-11ESect-10.2 P-12ESect-10.2 P-13ESect-10.2 P-14ESect-10.2 P-15ESect-10.2 P-16ESect-10.2 P-17ESect-10.2 P-18ESect-10.2 P-19ESect-10.2 P-20ESect-10.2 P-21ESect-10.2 P-22ESect-10.2 P-23ESect-10.2 P-24ESect-10.2 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