Answer
Total length: $s \simeq 30.50$
Work Step by Step
We have
$r = f\left( \theta \right) = \sqrt \theta $, ${\ \ }$ $f'\left( \theta \right) = \frac{1}{{2\sqrt \theta }}$.
Using Eq. (7) the total length of the curve is
$s = \mathop \smallint \limits_0^{4\pi } \sqrt {{{\left( {\sqrt \theta } \right)}^2} + {{\left( {\frac{1}{{2\sqrt \theta }}} \right)}^2}} {\rm{d}}\theta = \mathop \smallint \limits_0^{4\pi } \sqrt {\theta + \frac{1}{{4\theta }}} {\rm{d}}\theta $
Evaluating it using a computer algebra system we obtain the result:
$s \simeq 30.50$
