   Chapter 10.4, Problem 12E

Chapter
Section
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=2−cosθ 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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