Answer
$$40$$
Work Step by Step
\begin{aligned}
s &=\int_{a}^{b} \sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}} d t \\
&=\int_{0}^{2 \pi} \sqrt{100 \sin ^{4} \frac{\theta}{2}+100 \sin ^{2} \frac{\theta}{2} \cos ^{2} \frac{\theta}{2}} d \theta \\
&=\int_{0}^{2 \pi} \sqrt{100 \sin ^{2} \frac{\theta}{2}\left[\sin ^{2} \frac{\theta}{2}+\cos ^{2} \frac{\theta}{2}\right]} d \theta \\
&=\int_{0}^{2 \pi} \sqrt{100 \sin ^{2} \frac{\theta}{2}} d \theta \\
&=\int_{0}^{2 \pi} 10 \sin \frac{\theta}{2} d \theta \\
&=\left.10 \cdot \frac{-\cos \frac{\theta}{2}}{1 / 2}\right|_{0} ^{2 \pi}\\
&= 40
\end{aligned}
