   Chapter 10.4, Problem 36E

Chapter
Section
Textbook Problem

Find the area between a large loop and the enclosed small loop of the curve r = 1 + 2 cos 3θ.

To determine

To Find: The area between a large loop and the enclosed small loop of the curve.

Explanation

Given:

The polar equation is as below.

r=1+2cos3θ (1)

Calculation:

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

r=1+2cos3θ

Substitute 0 for θ in the equation (1).

r=1+2cos3(0)=3

Calculate the value of x.

x=rcosθ

Substitute 3 for r and 0 for θ .

x=3×cos(0×π180)=3

Calculate the value of y.

y=rsinθ

Substitute 3 for r and 0 for θ .

y=3×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+2cos3θ x=rcosθ y=rsinθ 0.00 3.00 3.00 0.00 10.00 2.73 2.69 0.47 20.00 2.00 1.88 0.68 30.00 1.00 0.87 0.50 40.00 0.00 0.00 0.00 50.00 -0.73 -0.47 -0.56 60.00 -1.00 -0.50 -0.87 70.00 -0.73 -0.25 -0.69 80.00 0.00 0.00 0.00 90.00 1.00 0.00 1.00 100.00 2.00 -0.35 1.97 110.00 2.73 -0.93 2.57 120.00 3.00 -1.50 2.60 130.00 2.73 -1.76 2.09 140.00 2.00 -1.53 1.29 150.00 1.00 -0.87 0.50 160.00 0.00 0.00 0.00 170.00 -0.73 0.72 -0.13 180.00 -1.00 1.00 0.00 190.00 -0.73 0.72 0.13 200.00 0.00 0.00 0.00 210.00 1.00 -0.87 -0.50 220.00 2.00 -1.53 -1.29 230.00 2.73 -1.76 -2.09 240.00 3.00 -1.50 -2.60 250.00 2.73 -0.93 -2.57 260.00 2.00 -0.35 -1.97 270.00 1.00 0.00 -1.00 280.00 0.00 0.00 0.00 290.00 -0.73 -0.25 0.69 300.00 -1.00 -0.50 0.87 310.00 -0.73 -0.47 0.56 320.00 0.00 0.00 0.00 330.00 1.00 0.87 -0.50 340.00 2.00 1.88 -0.68 350.00 2.73 2.69 -0.47 360.00 3.00 3.00 0.00

Graph:

The graph is plotted for x and y using the table (1) shown in figure (1).

Calculate the value of θ .

Substitute 0 for r in the equation (1).

1+2cos3θ=02cos3θ=1cos3θ=12

3θ=cos1(12)3θ=2π3,4π3θ=2π9,4π9

From figure (1), the region between the values 0 to 2π9 be half the large loop on the right, so we can double the integral over this interval for the area.

From 2π9 to 4π9 it makes a full small loop (lower left). Therefore, this is the interval for the smaller loop area integral.

Calculate the area of the region using the polar area formula.

A=2(12abr2dθ)largeloop(12abr2dθ)smallloop

Substitute (1+2cos3θ) for r in the above equation.

A=[(02π/9(1+2cos3θ)2dθ)12(2π/94π/9(1+2cos3θ)2dθ)]=[(02π/9(12+(2cos3θ)2+2(1)(2cos3θ))dθ)12(2π/94π/9(12+(2cos3θ)2+2(1)(2cos3θ))dθ)]=[(02π/9(1+4cos23θ+4cos3θ)dθ)12(2π/94π/9(1+4cos23θ+4cos3θ)dθ)]

Substitute 1+cos2θ2 for cos2θ in the above equation

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