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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the area of the region that lies inside the first curve and outside the second curve.

23. r = 4 sin θ, r = 2

To determine

To find: The area of the region that lies inside the first curve and outside the second curve.

Explanation

Given:

The polar equation is as below.

r=4sinθ (1)

r=2 (2)

Calculation:

Calculate the value of r in the equation (1).

r=4sinθ

Substitute 0° for θ in the equation (1).

r=4sin(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=4sinθ x=rcosθ y=rsinθ
0.00 0.00 0.00 0.00
10.00 0.69 0.68 0.12
20.00 1.37 1.29 0.47
30.00 2.00 1.73 1.00
40.00 2.57 1.97 1.65
50.00 3.06 1.97 2.35
60.00 3.46 1.73 3.00
70.00 3.76 1.29 3.53
80.00 3.94 0.68 3.88
90.00 4.00 0.00 4.00
100.00 3.94 -0.68 3.88
110.00 3.76 -1.29 3.53
120.00 3.46 -1.73 3.00
130.00 3.06 -1.97 2.35
140.00 2.57 -1.97 1.65
150.00 2.00 -1.73 1.00
160.00 1.37 -1.29 0.47
170.00 0.69 -0.68 0.12
180.00 0.00 0.00 0.00
190.00 -0.69 0.68 0.12
200.00 -1.37 1.29 0.47
210.00 -2.00 1.73 1.00
220.00 -2.57 1.97 1.65
230.00 -3.06 1.97 2.35
240.00 -3.46 1.73 3.00
250.00 -3.76 1.29 3.53
260.00 -3.94 0.68 3.88
270.00 -4.00 0.00 4.00
280.00 -3.94 -0.68 3.88
290.00 -3.76 -1.29 3.53
300.00 -3.46 -1.73 3.00
310.00 -3.06 -1.97 2.35
320.00 -2.57 -1.97 1.65
330.00 -2.00 -1.73 1.00
340.00 -1.37 -1.29 0.47
350.00 -0.69 -0.68 0.12
360.00 0.00 0.00 0.00

Calculate the value of x using the equation (2).

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==2×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 Radius(r) x=rcosθ y=rsinθ
0.00 2 2.00 0.00
10.00 2 1.97 0.35
20.00 2 1.88 0.68
30.00 2 1.73 1.00
40.00 2 1.53 1.29
50.00 2 1.29 1.53
60.00 2 1.00 1.73
70.00 2 0.68 1.88
80.00 2 0.35 1.97
90.00 2 0.00 2.00
100.00 2 -0.35 1.97
110.00 2 -0.68 1.88
120.00 2 -1.00 1.73
130.00 2 -1.29 1.53
140.00 2 -1.53 1.29
150.00 2 -1.73 1.00
160.00 2 -1.88 0.68
170.00 2 -1.97 0.35
180.00 2 -2.00 0.00
190.00 2 -1.97 -0.35
200.00 2 -1.88 -0.68
210.00 2 -1.73 -1.00
220.00 2 -1.53 -1.29
230.00 2 -1.29 -1.53
240.00 2 -1.00 -1.73
250.00 2 -0.68 -1.88
260.00 2 -0.35 -1.97
270.00 2 0.00 -2.00
280.00 2 0.35 -1.97
290

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