Chapter 10, Problem 32RE

### Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

Chapter
Section

### Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

# Find the area enclosed by the inner loop of the curve r = 1 − 3 sin θ.

To determine

To find: The area of the region that is enclosed within in the curve r=13sinθ.

Explanation

Given:

The polar equation of the curve with variable r is as follows. ,

r=13sinθ (1)

Assume θ=0

Calculate the value of r.

Substitute 0 for θ.in equation (1)

r=13sin(0×π180)r=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 (1).

 theta radius x y 0.00 1.00 1.00 0.00 10.00 0.48 0.47 0.08 20.00 -0.03 -0.02 -0.01 30.00 -0.50 -0.43 -0.25 40.00 -0.93 -0.71 -0.60 50.00 -1.30 -0.83 -0.99 60.00 -1.60 -0.80 -1.38 70.00 -1.82 -0.62 -1.71 80.00 -1.95 -0.34 -1.92 90.00 -2.00 0.00 -2.00 100.00 -1.95 0.34 -1.92 110.00 -1.82 0.62 -1.71 120.00 -1.60 0.80 -1.38 130.00 -1.30 0.83 -0.99 140.00 -0.93 0.71 -0.60 150.00 -0.50 0.43 -0.25 160.00 -0.03 0.02 -0.01 170.00 0.48 -0.47 0.08 180.00 1.00 -1.00 0.00 190.00 1.52 -1.50 -0.26 200.00 2.03 -1.90 -0.69 210.00 2.50 -2.17 -1.25 220.00 2.93 -2.24 -1.88 230.00 3.30 -2.12 -2.53 240.00 3.60 -1.80 -3.12 250.00 3.82 -1.31 -3.59 260.00 3.95 -0.69 -3.89 270.00 4.00 0.00 -4.00 280.00 3.95 0.69 -3.89 290.00 3.82 1.31 -3.59 300.00 3.60 1.80 -3.12 310.00 3.30 2.12 -2.53 320.00 2.93 2.24 -1.88 330.00 2.50 2.17 -1.25 340.00 2.03 1.90 -0.69 350.00 1.52 1.50 -0.26 360.00 1.00 1.00 0.00

Graph:

The graph is plotted for x and y using the table (1) in figure (1).

Determine the value of θ.

0=13sinθθ=sin1(13)

Two solutions are available between 0 and 2π they are θ and πθ.

So the limits are between sin1(13) and πsin1(13)

Calculate the area of the region using the polar area formula

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