Question

Prove that the area enclosed by all of the petals of the polar rose r = cos 2nθ is the same for every positive integer n.

Short answer

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

Step-by-step solution

Step 1. Given information

Equation of petal rose is$r=\cos 2n\theta$

Step 2. Explanation

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

$$\begin{gathered} =n{\int }_0^{\frac{2\mathrm{\pi }}{n}}\frac{1+\cos 4n\theta }{2}d\theta \\ =n\left[\frac{2\mathrm{\pi }}{2n}+\frac{\sin 4n\left({\displaystyle \frac{2\mathrm{\pi }}{n}}\right)}{2\left(4n\right)}-0\right] \\ =n\left(\frac{2\mathrm{\pi }}{2n}+0\right) \\ =\mathrm{\pi } \end{gathered}$$

So, the area enclosed by all of the petals of the polar rose r = cos 2nθ is the same for every positive integer n.