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

Question

Question:

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

Answer:

The areas of both regions are π2.

The graph of r=sin(2θ), 0≤θ<2π looks like this:
Since the area element in polar coordinates is rdrdθ, we can find the area of the four leaves above by
A=∫2π0∫sin(2θ)0rdrdθ.

Let us evaluate the inside integral first,
A=∫2π0[r2/2] sin(2θ)0dθ=∫2π0sin2(2θ)/2 x dθ
By the double-angle identity sin2(2θ)=1−cos(4θ)/2,
A=14∫2π0[1−cos(4θ)]dθ=1/4[θ−sin(4θ)/4]2π0=14[2π−sin(8π)/4−(0−sin(0)/4)]=1/4(2π)=π/2

Hence, the area is π2.
For r=cos(2θ), the area can be found by
A=∫2π0∫cos(2θ)0rdrdθ

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Sam Professor 2 years 1 Answer 1153 views 0

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