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Prove that the area enclosed by all of the petals of the polar rose r=sin2n+1θis the same for every positive integer n.

Short Answer

Expert verified

It is proved that the area enclosed by all of the petals of the given polar rose is the same for every positive integer n.

Step by step solution

01

Step 1. Given information

An equation of a rose petal isr=sin2n+1θ

02

Step 2. Explanation

Area =2n+11202π2n+1sin2n+1θ2dθ

=2n+11202π2n+11-cos2n+1θ2dθ=2n+14θ-sin2n+1θ2n+102π2n+1=π2

So, the area enclosed by all of the petals of the polar rose r=sin(2n+1)θis the same for every positive integer n.

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