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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the area of the region enclosed by one loop of the curve.

17. r = 4 cos 3θ

To determine

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

Explanation

Given:

The polar equation is r=4cos3θ .

Assume θ=0 .

Calculate the value of r.

r=4cos3θ

Substitute 0 for θ in the above equation.

r=4cos(0×π180)=4

Calculate the value of x.

x=rcosθ

Substitute 4 for r and 0 for θ in the above equation.

x=rcosθ=4×cos(0×π180)=4

Calculate the value of y.

y=rsinθ

Substitute 4 for r and 0 for θ in the above equation.

y=4×sin(0×π180)=0

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

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

θ r=4cos3θ x=rcosθ y=rsinθ
0.00 4.00 4.00 0.00
10.00 3.46 3.41 0.60
20.00 2.00 1.88 0.68
30.00 0.00 0.00 0.00
40.00 -2.00 -1.53 -1.29
50.00 -3.46 -2.23 -2.65
60.00 -4.00 -2.00 -3.46
70.00 -3.46 -1.18 -3.26
80.00 -2.00 -0.35 -1.97
90.00 0.00 0.00 0.00
100.00 2.00 -0.35 1.97
110.00 3.46 -1.18 3.26
120.00 4.00 -2.00 3.46
130.00 3.46 -2.23 2.65
140.00 2.00 -1.53 1.29
150.00 0.00 0.00 0.00
160.00 -2.00 1.88 -0.68
170.00 -3.46 3.41 -0.60
180.00 -4.00 4.00 0.00

Graph:

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

Find the limit of integration for two consecutive value of θ for which r is zero.

From the figure (1), the two consecutive value for which 4cos3θ is zero is -π6 to π6 .

Therefore, integrate either -π6 to π6 or π6 to π2 .

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

A=12abr2dθ (1)

Substitute 4cos3θ for r in the equation (1)

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Chapter 10 Solutions

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P-52RECh-10 P-53RECh-10 P-54RECh-10 P-55RECh-10 P-56RECh-10 P-57RECh-10 P-58RECh-10 P-1PCh-10 P-2PCh-10 P-3PCh-10 P-4PCh-10 P-5PCh-10 P-6P

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