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

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Answer 1 · 2014-11-02T12:59:43

Let us look at the region.

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(Note: #r^2=cos2theta# in purple, and #r^2=sin2theta# in blue)

Since there two identical region, we will find a half of one region then multiply by #4#. The combine area #A# can be found by

#A=4int_0^{pi/8}int_0^{sqrt{2sin2theta}}rdrd theta#

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

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

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

#=-cos(pi/4)+cos(0)#

#=-{sqrt{2}}/{2}+1={2-sqrt{2}}/2#

I hope that this was helpful.