   Chapter 10.4, Problem 9E

Chapter
Section
Textbook Problem

Sketch the curve and find the area that it encloses.9. r = 2 sin θ

To determine

To find: The area of the region that the polar equation encloses.

Explanation

Given:

The polar equation is given as below.

r=2sinθ (1)

Assume the value of θ=180 .

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

r=2sinθ

Substitute 180 for θ in the equation (1).

r=2sin(180×π180)=0

Calculate the value of x.

x=rcosθ

Substitute 0 for r and 180 for θ .

x=rcosθ=0×cos(180×π180)=0

Calculate the value of y.

y=rsinθ

Substitute 0 for r and 180 for θ .

y=0×sin(180×π180)=0

Similarly, calculate the values of x and y using the value of θ from 0 to 180 .

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

 θ r=2sinθ x=rcosθ y=rsinθ 0.0 0.0 0.0 0.0 10.0 0.3 0.3 0.1 20.0 0.7 0.6 0.2 30.0 1.0 0.9 0.5 40.0 1.3 1.0 0.8 50.0 1.5 1.0 1.2 60.0 1.7 0.9 1.5 70.0 1.9 0.6 1.8 80.0 2.0 0.3 1.9 90.0 2.0 0.0 2.0 100.0 2.0 -0.3 1.9 110.0 1.9 -0.6 1.8 120.0 1.7 -0.9 1.5 130.0 1.5 -1.0 1.2 140.0 1.3 -1.0 0.8 150.0 1.0 -0.9 0.5 160.0 0.7 -0.6 0.2 170.0 0.3 -0.3 0.1 180.0 0.0 0.0 0.0

Graph:

The graph is plotted between x and y as shown in figure (1)

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