Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.2 Arc Length and Speed - Exercises - Page 610: 8

$$2$$

\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*}