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

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1 Answer
Wataru
Nov 9, 2014

The region bounded by the polar curves looks like:

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Since the region consists of two identical leaves that are symmetric about the #y#-axis, I will try to find a half of one leaf then multiply it by #4#.

#A=4int_0^{pi/2}int_0^{1-cos theta}rdrd theta#

#=4int_0^{pi/2}[r^2/2]_0^{1-cos theta}d theta#

#=2int_0^{pi/2}(1-2cos theta+cos^2theta)d theta#

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

#=int_0^{pi/2}(3-4cos theta+cos2theta)d theta#

#=[3theta-4sintheta+1/2sin2theta]_0^{pi/2}#

#={3pi}/2-4#

I hope that this was helpful.

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