Answer
total length $\simeq 15.40$
Work Step by Step
$r = f\left( \theta \right) = 2 + \sin 2\theta $, ${\ \ }$ $f'\left( \theta \right) = 2\cos 2\theta $.
From Exercise 19 and 20 we see that the entire curve is in the interval $0 \le \theta \le 2\pi $. So, using Eq. (7) the total length of the curve is
$s = \mathop \smallint \limits_0^{2\pi } \sqrt {{{\left( {2 + \sin 2\theta } \right)}^2} + {{\left( {2\cos 2\theta } \right)}^2}} {\rm{d}}\theta $
Evaluating it using a computer algebra system we obtain the result:
$s \simeq 15.40$
