Answer
$$8\left(-\frac{1}{\sqrt{2}}+1\right)$$
Work Step by Step
\begin{aligned}
s &=\int_{a}^{b} \sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}} d t \\
&=\int_{0}^{\frac{x}{2}} \sqrt{4(\sin 2 t-\sin t)^{2}+4(\cos t-\cos 2 t)^{2}} d t \\
&=\int_{0}^{\frac{\pi}{2}} 2 \sqrt{\sin ^{2}(2 t)-2 \sin (2 t) \sin t+\sin ^{2} t+\cos ^{2} t-2 \cos (2 t) \cos t+\cos ^{2} t} d t \\
&=\int_{0}^{\frac{\pi}{2}} 2 \sqrt{\sin ^{2}(2 t)-2 \sin (2 t) \sin t+\cos ^{2} t-2 \cos (2 t) \cos t+1} d t \\
&= \int_{0}^{\frac{\pi}{2}} 2 \sqrt{\sin ^{2}(2 t)-2 \sin (2 t) \sin t+\cos ^{2} t-2 \cos (2 t) \cos t+1} d t \\
&=2 \int_{0}^{\frac{\pi}{2}} \sqrt{-2 \sin (2 t) \sin t-2 \cos (2 t) \cos t+2} d t \\
&=2 \int_{0}^{\frac{\pi}{2}} \sqrt{2} \sqrt{-2 \sin t \cos t \sin t-\left(1-2 \sin ^{2} t\right)\left(1-2 \sin ^{2} \frac{t}{2}\right)+1} d t\\
&=2 \sqrt{2} \int_{0}^{\frac{\pi}{2}} \sqrt{-2 \sin ^{2} t\left(1-2 \sin ^{2} \frac{t}{2}\right)-\left(1-2 \sin ^{2} t-2 \sin ^{2} \frac{t}{2}+4 \sin ^{2} t \sin ^{2} \frac{t}{2}\right)+1} d t\\
&=2 \sqrt{2} \int_{0}^{\frac{\pi}{2}} \sqrt{2} \sin \frac{t}{2} d t\\
&=4 \int_{0}^{\frac{\pi}{2}} \sin \frac{t}{2} d t\\
&=\left.4 \cdot \frac{-\cos \frac{t}{2}}{1 / 2}\right|_{0} ^{\frac{\pi}{2}}
\\
&=8\left(-\frac{1}{\sqrt{2}}+1\right)
\end{aligned}
