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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

Sketch the curve and find the area that it encloses.

12. r = 2 − cos θ

To determine

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

Explanation

The polar equation is r=2cosθ .

Assume the value of θ=180° .

Calculate the value of r.

r=2cosθ=2cos(180×π180)=3

Calculate the value of x coordinate of curve.

x=rcosθ

Substitute 3 for r and 180° for θ .

x=rcosθ=3cos(180×π180)=3

Calculate the value of y coordinate of curve.

y=rsinθ

Substitute 3 for r and 180° for θ .

y=3sin(180×π180)=0

Repeat the calculation of the values of x and y using the value of θ from 180° to 180° .

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

θ r=2cosθ x=rcosθ y=rsinθ
−180 3 −3 0
−170 2.984808 −2.93946 −0.51831
−160 2.939693 −2.76241 −1.00543
−150 2.866025 −2.48205 −1.43301
−140 2.766044 −2.11891 −1.77798
−130 2.642788 −1.69875 −2.02449
−120 2.5 −1.25 −2.16506
−110 2.34202 −0.80102 −2.20078
−100 2.173648 −0.37745 −2.14063
−90 2 1.23E−16 −2
−80 1.826352 0.317143 −1.79861
−70 1.65798 0.567063 −1.55799
−60 1.5 0.75 −1.29904
−50 1.357212 0.872399 −1.03969
−40 1.233956 0.945265 −0.79317
−30 1.133975 0.982051 −0.56699
−20 1.060307 0.996363 −0.36265
−10 1.015192 0.999769 −0.17629
0 1 1 0
10 1.015192 0.999769 0.176286
20 1.060307 0.996363 0.362646
30 1.133975 0.982051 0.566987
40 1.233956 0.945265 0.793171
50 1.357212 0.872399 1.039685
60 1.5 0.75 1.299038
70 1.65798 0.567063 1.557991
80 1.826352 0.317143 1.798605
90 2 1.23E−16 2
100 2.173648 −0.37745 2.140626
110 2.34202 −0.80102 2.200779
120 2.5 −1.25 2.165064
130 2.642788 −1.69875 2.024493
140 2.766044 −2.11891 1.777979
150 2.866025 −2.48205 1.433013
160 2.939693 −2.76241 1.005434
170 2.984808 −2.93946 0.518306
180 3 −3 0

Plot the graph as below.

Refer the graph. The curve lies on all the quadrant and the limit tends to be between 0θ2π .

Calculate the area of the region using the polar area formula

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