Question

Prove that the area enclosed by all of the petals of the polar rose $r=\sin \left(2n+1\right)\theta$is the same for every positive integer n.

Short answer

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

Step 1. Given information

An equation of a rose petal is$r=\sin \left(2n+1\right)\theta$

Step 2. Explanation

Area $=\left(2n+1\right)\left(\frac{1}{2}\right){\int }_0^{\frac{2\mathrm{\pi }}{2n+1}}{\left(\sin \left(2n+1\right)\theta \right)}^{2}d\theta$

$$\begin{gathered} =\left(2n+1\right)\left(\frac{1}{2}\right){\int }_0^{\frac{2\mathrm{\pi }}{2n+1}}\frac{1-\cos \left(2n+1\right)\theta }{2}d\theta \\ =\left(\frac{2n+1}{4}\right){\left[\theta -\frac{\sin \left(2n+1\right)\theta }{2n+1}\right]}_0^{\frac{2\mathrm{\pi }}{2n+1}} \\ =\frac{\mathrm{\pi }}{2} \end{gathered}$$

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