   Chapter 10.4, Problem 31E

Chapter
Section
Textbook Problem

Find the area of the region that lies inside both curves.31. r = sin 2θ, r = cos 2θ

To determine

To Find: The area of the region that lies inside both curves.

Explanation

Given:

The polar equations are as below.

r=sin2θ (1)

r=cos2θ (2)

Calculation:

Assume the value of θ=0 .

Calculate the value of r from equation (1).

r=sin2θ

Substitute 0 for θ .

r=sin2(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=sin2θ x=rcosθ y=rsinθ 0.00 0.00 0.00 0.00 10.00 0.34 0.34 0.06 20.00 0.64 0.60 0.22 30.00 0.87 0.75 0.43 40.00 0.98 0.75 0.63 50.00 0.98 0.63 0.75 60.00 0.87 0.43 0.75 70.00 0.64 0.22 0.60 80.00 0.34 0.06 0.34 90.00 0.00 0.00 0.00 100.00 -0.34 0.06 -0.34 110.00 -0.64 0.22 -0.60 120.00 -0.87 0.43 -0.75 130.00 -0.98 0.63 -0.75 140.00 -0.98 0.75 -0.63 150.00 -0.87 0.75 -0.43 160.00 -0.64 0.60 -0.22 170.00 -0.34 0.34 -0.06 180.00 0.00 0.00 0.00 190.00 0.34 -0.34 -0.06 200.00 0.64 -0.60 -0.22 210.00 0.87 -0.75 -0.43 220.00 0.98 -0.75 -0.63 230.00 0.98 -0.63 -0.75 240.00 0.87 -0.43 -0.75 250.00 0.64 -0.22 -0.60 260.00 0.34 -0.06 -0.34 270.00 0.00 0.00 0.00 280.00 -0.34 -0.06 0.34 290.00 -0.64 -0.22 0.60 300.00 -0.87 -0.43 0.75 310.00 -0.98 -0.63 0.75 320.00 -0.98 -0.75 0.63 330.00 -0.87 -0.75 0.43 340.00 -0.64 -0.60 0.22 350.00 -0.34 -0.34 0.06 360.00 0.00 0.00 0.00

Calculate the value of r from equation (2).

r=cos2θ

Substitute 0 for θ in the equation (2).

r=cos2(0×π180)=1

Calculate the value of x.

x=rcosθ

Substitute 1 for r and 0 for θ .

x=rcosθ=1×cos(0×π180)=1

Calculate the value of y.

y=rsinθ

Substitute 0 for r and 0 for θ .

y=1×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 (2).

 θ r=cos2θ x=rcosθ y=rsinθ 0.00 1.00 1.00 0.00 10.00 0.94 0.93 0.16 20.00 0.77 0.72 0.26 30.00 0.50 0.43 0.25 40.00 0.17 0.13 0.11 50.00 -0.17 -0.11 -0.13 60.00 -0.50 -0.25 -0.43 70.00 -0.77 -0

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