   Chapter 10.4, Problem 19E

Chapter
Section
Textbook Problem

# Find the area of the region enclosed by one loop of the curve.19. r = sin 4θ

To determine

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

Explanation

Given:

The polar equation is r=sin4θ .

Assume θ=0

Calculate the value of r.

r=sin4θ

Substitute 0 for θ in the above equation.

r=sin4(0×π180)=0

Calculate the value of x.

x=rcosθ

Substitute 0 for r and 0 for θ .

x=rcosθ=0×cos(0×π180)=0

Calculate the value of y.

y=rsinθ

Substitute 0 for r and 0 for θ .

y=0×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=sin4θ x=rcosθ y=rsinθ 0.00 0.00 0.00 0.00 1.00 0.07 0.07 0.00 20.00 0.98 0.93 0.34 30.00 0.87 0.75 0.43 40.00 0.34 0.26 0.22 50.00 -0.34 -0.22 -0.26 60.00 -0.87 -0.43 -0.75 70.00 -0.98 -0.34 -0.93 80.00 -0.64 -0.11 -0.63 90.00 0.00 0.00 0.00 100.00 0.64 -0.11 0.63 110.00 0.98 -0.34 0.93 120.00 0.87 -0.43 0.75 130.00 0.34 -0.22 0.26 140.00 -0.34 0.26 -0.22 150.00 -0.87 0.75 -0.43 160.00 -0.98 0.93 -0.34 170.00 -0.64 0.63 -0.11 180.00 0.00 0.00 0.00 190.00 0.64 -0.63 -0.11 200.00 0.98 -0.93 -0.34 210.00 0.87 -0.75 -0.43 220.00 0.34 -0.26 -0.22 230.00 -0.34 0.22 0.26 240.00 -0.87 0.43 0.75 250.00 -0.98 0.34 0.93 260.00 -0.64 0.11 0.63 270.00 0.00 0.00 0.00 280.00 0.64 0.11 -0.63 290.00 0.98 0.34 -0.93 300.00 0.87 0.43 -0.75 310.00 0.34 0.22 -0.26 320.00 -0.34 -0.26 0.22 330.00 -0.87 -0.75 0.43 340.00 -0.98 -0.93 0.34 350.00 -0.64 -0.63 0.11 360.00 0.00 0.00 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 sin4θ is zero is 0 to π4 .

Therefore, integrate either 0 to π4 or -π4 to 0 .

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

A=12abr2dθ (1)

Substitute sin4θ for r in the equation (1)

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