Answer
total length $\simeq 6.68$
Work Step by Step
The three-petal rose in Figure 20 has polar equation $r = \cos 3\theta $. So, we have
$r = f\left( \theta \right) = \cos 3\theta $, ${\ \ }$ $f'\left( \theta \right) = - 3\sin 3\theta $.
When $\theta=0$, we have $r=1$. So, the curve starts at $\left( {1,0} \right)$. When $\theta = \frac{\pi }{6}$, we have $r=0$. So, the curve reaches the origin when $\theta = \frac{\pi }{6}$. From Figure 20, we see that the curve completed the upper-half of the petal for the interval $0 \le \theta \le \frac{\pi }{6}$. By symmetry, the entire curve is in the interval $0 \le \theta \le \pi $. Using Eq. (7) the total length of the curve is
$s = \mathop \smallint \limits_0^\pi \sqrt {{{\cos }^2}3\theta + {{\left( { - 3\sin 3\theta } \right)}^2}} {\rm{d}}\theta $
$s = \mathop \smallint \limits_0^\pi \sqrt {{{\cos }^2}3\theta + 9{{\sin }^2}3\theta } {\rm{d}}\theta $
Evaluating it using a computer algebra system we obtain the result:
$s \simeq 6.68$
