#### Answer

$$2$$

#### Work Step by Step

\begin{align*}
s&=\int_{a}^{b} \sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}} d t\\
&=\int_{0}^{2} \sqrt{\theta^{2} \sin ^{2} \theta+\theta^{2} \cos ^{2} \theta} d \theta\\
&=\int_{0}^{2} \sqrt{\theta^{2}\left(\sin ^{2} \theta+\cos ^{2} \theta\right)} d \theta\\
&= \int_{0}^{2}\theta d \theta\\
&= \frac{1}{2}\theta^2\bigg|_{0}^{2}\\
&=2
\end{align*}