   Chapter 10.4, Problem 28E

Chapter
Section
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.

r=3sinθ (1)

r=2sinθ (2)

Calculation:

Assume the value of θ=0 .

Calculate the value of r using equation (1).

r=3sinθ

Substitute 0 for θ in the equation (1).

r=3sin(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).

 θ r=3sinθ x=rcosθ y=rsinθ 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), r=2sinθ

Substitute 0 for θ in the equation (2).

r=2sin(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 (2).

 θ r=2−sinθ x=rcosθ y=rsinθ 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

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