   Chapter 10.4, Problem 26E

Chapter
Section
Textbook Problem

Find the area of the region that lies inside the first curve and outside the second curve.26. r = 1 + cos θ, r = 2 − cos 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 equation is as below.

r=1+cosθ (1)

r=2cosθ (2)

Calculation:

Assume the value of θ=0.

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

r=1+cosθ

Substitute 0 for θ in the equation (1).

r=1+cos(0×π180)=2

Calculate the value of x.

x=rcosθ

Substitute 2 for r and 0 for θ.

x=rcosθ=2×cos(0×π180)=2

Calculate the value of y.

y=rsinθ

Substitute 2 for r and 0 for θ.

y=2×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).

 θ r=1+cosθ x=rcosθ y=rsinθ 0.00 2.00 2.00 0.00 10.00 1.98 1.95 0.34 20.00 1.94 1.82 0.66 30.00 1.87 1.62 0.93 40.00 1.77 1.35 1.14 50.00 1.64 1.06 1.26 60.00 1.50 0.75 1.30 70.00 1.34 0.46 1.26 80.00 1.17 0.20 1.16 90.00 1.00 0.00 1.00 100.00 0.83 -0.14 0.81 110.00 0.66 -0.23 0.62 120.00 0.50 -0.25 0.43 130.00 0.36 -0.23 0.27 140.00 0.23 -0.18 0.15 150.00 0.13 -0.12 0.07 160.00 0.06 -0.06 0.02 170.00 0.02 -0.01 0.00 180.00 0.00 0.00 0.00 190.00 0.02 -0.01 0.00 200.00 0.06 -0.06 -0.02 210.00 0.13 -0.12 -0.07 220.00 0.23 -0.18 -0.15 230.00 0.36 -0.23 -0.27 240.00 0.50 -0.25 -0.43 250.00 0.66 -0.23 -0.62 260.00 0.83 -0.14 -0.81 270.00 1.00 0.00 -1.00 280.00 1.17 0.20 -1.16 290.00 1.34 0.46 -1.26 300.00 1.50 0.75 -1.30 310.00 1.64 1.06 -1.26 320.00 1.77 1.35 -1.14 330.00 1.87 1.62 -0.93 340.00 1.94 1.82 -0.66 350.00 1.98 1.95 -0.34 360.00 2.00 2.00 0.00

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

r=2cosθ

Substitute 0 for θ in the equation (2).

r=2cos(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=2×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).

 θ r=2−cosθ x=rcosθ y=rsinθ 0.00 1.00 1.00 0.00 10.00 1.02 1.00 0.18 20.00 1.06 1.00 0.36 30.00 1.13 0.98 0.57 40.00 1.23 0.95 0.79 50.00 1.36 0.87 1.04 60

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