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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
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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