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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.

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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P-20ESect-10.6 P-21ESect-10.6 P-22ESect-10.6 P-23ESect-10.6 P-24ESect-10.6 P-25ESect-10.6 P-26ESect-10.6 P-27ESect-10.6 P-28ESect-10.6 P-29ESect-10.6 P-30ESect-10.6 P-31ECh-10 P-1RCCCh-10 P-2RCCCh-10 P-3RCCCh-10 P-4RCCCh-10 P-5RCCCh-10 P-6RCCCh-10 P-7RCCCh-10 P-8RCCCh-10 P-9RCCCh-10 P-1RQCh-10 P-2RQCh-10 P-3RQCh-10 P-4RQCh-10 P-5RQCh-10 P-6RQCh-10 P-7RQCh-10 P-8RQCh-10 P-9RQCh-10 P-10RQCh-10 P-1RECh-10 P-2RECh-10 P-3RECh-10 P-4RECh-10 P-5RECh-10 P-6RECh-10 P-7RECh-10 P-8RECh-10 P-9RECh-10 P-10RECh-10 P-11RECh-10 P-12RECh-10 P-13RECh-10 P-14RECh-10 P-15RECh-10 P-16RECh-10 P-17RECh-10 P-18RECh-10 P-19RECh-10 P-20RECh-10 P-21RECh-10 P-22RECh-10 P-23RECh-10 P-24RECh-10 P-25RECh-10 P-26RECh-10 P-27RECh-10 P-28RECh-10 P-29RECh-10 P-30RECh-10 P-31RECh-10 P-32RECh-10 P-33RECh-10 P-34RECh-10 P-35RECh-10 P-36RECh-10 P-37RECh-10 P-38RECh-10 P-39RECh-10 P-40RECh-10 P-41RECh-10 P-42RECh-10 P-43RECh-10 P-44RECh-10 P-45RECh-10 P-46RECh-10 P-47RECh-10 P-48RECh-10 P-49RECh-10 P-50RECh-10 P-51RECh-10 P-52RECh-10 P-53RECh-10 P-54RECh-10 P-55RECh-10 P-56RECh-10 P-57RECh-10 P-58RECh-10 P-1PCh-10 P-2PCh-10 P-3PCh-10 P-4PCh-10 P-5PCh-10 P-6P

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