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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.

18. r2 = 4 cos 2θ

To determine

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

Explanation

Given:

The polar equation is r2=4cos2θ .

Assume θ=0

Calculate the value of r.

r=4cos2θ

Substitute 0 for θ in the above equation.

r=4cos2θ=4cos2(0×π180)=2

Calculate the value of x.

x=rcosθ

Substitute 2 for r and 0 for θ .

x=rcosθ=2×cos(0×π180)=2

Calculate the value of y.

y=rsinθ

Substitute 2 for r and 0 for θ .

y=4×sin(0×π180)=0

Similarly calculate the values of x and y using the value of θ from 45 to 45 and 225 to 225 .

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

θ r=4cos2θ x=rcosθ y=rsinθ
-45.00 0.00 0.00 0.00
-44.00 0.37 0.27 -0.26
-43.00 0.53 0.39 -0.36
-42.00 0.65 0.48 -0.43
-41.00 0.75 0.56 -0.49
-40.00 0.83 0.64 -0.54
-35.00 1.17 0.96 -0.67
-30.00 1.41 1.22 -0.71
-25.00 1.60 1.45 -0.68
-20.00 1.75 1.64 -0.60
-15.00 1.86 1.80 -0.48
-10.00 1.94 1.91 -0.34
-5.00 1.98 1.98 -0.17
0.00 2.00 2.00 0.00
5.00 1.98 1.98 0.17
10.00 1.94 1.91 0.34
15.00 1.86 1.80 0.48
20.00 1.75 1.64 0.60
25.00 1.60 1.45 0.68
30.00 1.41 1.22 0.71
35.00 1.17 0.96 0.67
40.00 0.83 0.64 0.54
41.00 0.75 0.56 0.49
42.00 0.65 0.48 0.43
43.00 0.53 0.39 0.36
44.00 0.37 0.27 0.26
45.00 0.00 0.00 0.00
-225.00 0.00 0.00 0.00
-220.00 0.83 -0.64 0.54
-215.00 1.17 -0.96 0.67
-210.00 1.41 -1.22 0.71
-205.00 1.60 -1.45 0.68
-200

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