How do you find the area of the region bounded by the polar curves r^2=cos(2theta)r2=cos(2θ) and r^2=sin(2theta)r2=sin(2θ) ?

1 Answer
Nov 2, 2014

Let us look at the region.

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(Note: r^2=cos2thetar2=cos2θ in purple, and r^2=sin2thetar2=sin2θ in blue)

Since there two identical region, we will find a half of one region then multiply by 44. The combine area AA can be found by

A=4int_0^{pi/8}int_0^{sqrt{2sin2theta}}rdrd thetaA=4π802sin2θ0rdrdθ

=4int_0^{pi/8}[r^2/2]_0^{sqrt{sin2theta}}d theta=4π80[r22]sin2θ0dθ

=2 int_0^{pi/8}sin2theta d theta=2π80sin2θdθ

=2[{-cos2theta}/2]_0^{pi/8}=2[cos2θ2]π80

=-cos(pi/4)+cos(0)=cos(π4)+cos(0)

=-{sqrt{2}}/{2}+1={2-sqrt{2}}/2=22+1=222


I hope that this was helpful.