Answer
Total length: $s \simeq 79.56$
Work Step by Step
We have
$r = f\left( \theta \right) = \theta \sin \theta $, ${\ \ }$ $f'\left( \theta \right) = \sin \theta + \theta \cos \theta $.
Using Eq. (7) the total length of the curve is
$s = \mathop \smallint \limits_0^{4\pi } \sqrt {{{\left( {\theta \sin \theta } \right)}^2} + {{\left( {\sin \theta + \theta \cos \theta } \right)}^2}} {\rm{d}}\theta $
Evaluating it using a computer algebra system we obtain the result:
$s \simeq 79.56$
