Chapter 10.4, Problem 17E

### Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

Chapter
Section

### 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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