Chapter 10.3, Problem 23ES

(a) Show that the arc length of one petal of the rose $r=\cos n\theta$ is given by

$2{\displaystyle {\int }_0^{\pi /\left(2n\right)}\sqrt{1+\left(n^2-1\right){\sin }^{2}n\theta d\theta }}$

(b) Use the numerical integration capability of a calculating utility to approximate the arc length of one petal of the four-petal rose $r=\cos 2\theta .$

(c) Use the numerical integration capability of a calculating utility to approximate the arc length of one petal of the n-petal rose $r=\cos n\theta \text{for}n=2,3,4,\dots ,20;$ then make a conjecture about the limit of these arc lengths as $n\to +\infty .$