How do you find the area of the region bounded by the polar curves r=1+cos(theta)r=1+cos(θ) and r=1-cos(theta)r=1−cos(θ) ?

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1 Answer
Wataru
Nov 9, 2014

The region bounded by the polar curves looks like:

enter image source here

Since the region consists of two identical leaves that are symmetric about the yy-axis, I will try to find a half of one leaf then multiply it by 44.

A=4int_0^{pi/2}int_0^{1-cos theta}rdrd thetaA=4∫π20∫1−cosθ0rdrdθ

=4int_0^{pi/2}[r^2/2]_0^{1-cos theta}d theta=4∫π20[r22]1−cosθ0dθ

=2int_0^{pi/2}(1-2cos theta+cos^2theta)d theta=2∫π20(1−2cosθ+cos2θ)dθ

by cos^2theta=1/2(1+cos2theta)cos2θ=12(1+cos2θ),

=int_0^{pi/2}(3-4cos theta+cos2theta)d theta=∫π20(3−4cosθ+cos2θ)dθ

=[3theta-4sintheta+1/2sin2theta]_0^{pi/2}=[3θ−4sinθ+12sin2θ]π20

={3pi}/2-4=3π2−4

I hope that this was helpful.

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