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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the area of the region that lies inside both of the circles r = 2 sin θ and r = sin θ + cos θ.

To determine

To find: The area of the region that lies inside both curves r=2sinθ and r=sinθ+cosθ.

Explanation

Given:

The polar equations are

r=2sinθ (1)

r=sinθ+cosθ (2)

Calculation:

Calculate the value of r from the equation (1).

r=2sinθ

Substitute 0 for θ in the equation (1).

r=2×sin(0×π180)=0

Calculate the value of x.

x=rcosθ

Substitute 0 for r and 0 for θ.

x=rcosθ=0×cos(0×π180)=0

Calculate the value of y.

y=rsinθ

Substitute 0 for r and 0 for θ.

y=0×sin(0×π180)=0

Similarly calculate the values of x and y using the value of θ from 0 to 360

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

theta radius x y
0.00 0.00 0.00 0.00
10.00 0.35 0.34 0.06
20.00 0.68 0.64 0.23
30.00 1.00 0.87 0.50
40.00 1.29 0.98 0.83
50.00 1.53 0.98 1.17
60.00 1.73 0.87 1.50
70.00 1.88 0.64 1.77
80.00 1.97 0.34 1.94
90.00 2.00 0.00 2.00
100.00 1.97 -0.34 1.94
110.00 1.88 -0.64 1.77
120.00 1.73 -0.87 1.50
130.00 1.53 -0.98 1.17
140.00 1.29 -0.98 0.83
150.00 1.00 -0.87 0.50
160.00 0.68 -0.64 0.23
170.00 0.35 -0.34 0.06
180.00 0.00 0.00 0.00
190.00 -0.35 0.34 0.06
200.00 -0.68 0.64 0.23
210.00 -1.00 0.87 0.50
220.00 -1.29 0.98 0.83
230.00 -1.53 0.98 1.17
240.00 -1.73 0.87 1.50
250.00 -1.88 0.64 1.77
260.00 -1.97 0.34 1.94
270.00 -2.00 0.00 2.00
280.00 -1.97 -0.34 1.94
290.00 -1.88 -0.64 1.77
300.00 -1.73 -0.87 1.50
310.00 -1.53 -0.98 1.17
320.00 -1.29 -0.98 0.83
330.00 -1.00 -0.87 0.50
340.00 -0.68 -0.64 0.23
350.00 -0.35 -0.34 0.06
360.00 0.00 0.00 0.00

Calculate the value of r from the equation (2).

r=sinθ+cosθ

Substitute 0 for θ in the equation (2).

r=sin(0×π180)+cos(0×π180)=1

Calculate the value of x.

x=rcosθ

Substitute 1 for r and 0 for θ.

x=rcosθ=1×cos(0×π180)=1

Calculate the value of y.

y=rsinθ

Substitute 1 for r and 0 for θ.

y=1×sin(0×π180)=0

Similarly calculate the values of x and y using the value of θ from 0 to 360.

Tabulate the values of x and y in table (2).

theta radius x y
0.00 1.00 1.00 0.00
10.00 1.16 1.14 0.20
20.00 1.28 1.20 0.44
30.00 1.37 1.18 0.68
40.00 1.41 1.08 0.91
50.00 1.41 0.91 1.08
60.00 1.37 0.68 1.18
70.00 1.28 0.44 1.20
80.00 1.16 0.20 1.14
90.00 1.00 0.00 1.00
100.00 0.81 -0.14 0.80
110.00 0.60 -0.20 0.56
120.00 0.37 -0.18 0.32
130.00 0.12 -0.08 0.09
140.00 -0.12 0.09 -0.08
150.00 -0.37 0.32 -0.18
160.00 -0.60 0.56 -0.20
170.00 -0.81 0.80 -0.14
180.00 -1.00 1.00 0.00
190.00 -1.16 1.14 0.20
200.00 -1.28 1.20 0.44
210.00 -1.37 1.18 0.68
220.00 -1.41 1.08 0.91
230.00 -1.41 0.91 1.08
240.00 -1.37 0.68 1.18
250.00 -1.28 0.44 1.20
260.00 -1.16 0.20 1.14
270.00 -1.00 0.00 1.00
280.00 -0.81 -0.14 0.80
290.00 -0.60 -0.20 0.56
300.00 -0.37 -0.18 0.32
310.00 -0.12 -0.08 0.09
320.00 0.12 0.09 -0.08
330.00 0.37 0.32 -0.18
340.00 0.60 0.56 -0.20
350.00 0.81 0.80 -0

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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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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 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