Answer
The length of the curve is given by:
$L\approx 17.1568$
Work Step by Step
The curve with polar equation
$$
r=\sin(\frac{\theta}{4})
$$
is completely traced with $ 0 \leq \theta \leq 8\pi $.
The length of the curve is given by the following:
$$
\begin{split}
L=& \int_{0}^{8\pi } \sqrt{r^{2}+\left(\frac{d r}{d \theta}\right)^{2}} d \theta\\
&=\int_{0}^{8\pi } \sqrt{(\sin(\frac{\theta}{4}) )^{2} +(\frac{1}{4}\cos(\theta{4}))^{2}} d \theta\\
&=\int_{0}^{8\pi } \sqrt{(\sin(\frac{\theta}{4}) )^{2} +\frac{1}{16}(\cos(\theta{4}))^{2}} d \theta\\
&\approx 17.1568
\end{split}
$$
