   Chapter 10, Problem 31RE

Chapter
Section
Textbook Problem

Find the area enclosed by the curve r2 = 9 cos 5θ.

To determine

To find: The area enclosed within in the curve r2=9cos5θ.

Explanation

Given:

The polar equation of the curve with variable r is as follows. ,

r2=9cos5θ

r=±9cos5θ (1)

Assume θ=0°.

Calculate the value of r.

Substitute 0° for θ.in equation (1)

r=13sin(0×π180)r=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 0 for r and 0° for θ.

y=1×sin(0×π180)=0

Similarly calculate the values of x and y and the negative values of x and y as x’ and y’ as the values of θ from 0° to 360° as shown in the table (1).

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

 theta radius x y x' y' 0.00 3.00 3.00 0.00 -3.00 0.00 10.00 2.41 2.37 0.42 -2.37 -0.42 20.00 ∞ ∞ ∞ ∞ ∞ 30.00 ∞ ∞ ∞ ∞ ∞ 40.00 ∞ ∞ ∞ ∞ ∞ 50.00 60.00 2.12 1.06 1.84 -1.06 -1.84 70.00 2.98 1.02 2.80 -1.02 -2.80 80.00 2.63 0.46 2.59 -0.46 -2.59 90.00 0.00 0.00 0.00 0.00 0.00 100.00 ∞ ∞ ∞ ∞ ∞ 110.00 ∞ ∞ ∞ ∞ ∞ 120.00 ∞ ∞ ∞ ∞ ∞ 130.00 1.75 -1.13 1.34 1.13 -1.34 140.00 2.91 -2.23 1.87 2.23 -1.87 150.00 2.79 -2.42 1.40 2.42 -1.40 160.00 1.25 -1.17 0.43 1.17 -0.43 170.00 ∞ ∞ ∞ ∞ ∞ 180.00 ∞ ∞ ∞ ∞ ∞ 190

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