   Chapter 10, Problem 35RE

Chapter
Section
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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