Answer
$$\frac{13\sqrt{13}-8}{27}$$
Work Step by Step
\begin{align*}
s&=\int_{a}^{b} \sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}} d t\\
&=\int_{0}^{1} \sqrt{9 t^{4}+4 t^{2}} d t\\
&=\int_{0}^{1} \sqrt{t^{2}\left(9 t^{2}+4\right)} d t\\
&=\frac{1}{18} \int_{0}^{1} 18 t\left(9 t^{2}+4\right)^{1 / 2} d t\\
&=\left.\frac{1}{18} \frac{\left(9 t^{2}+4\right)^{3 / 2}}{3 / 2}\right|_{0} ^{1}\\
&=\frac{13\sqrt{13}-8}{27}
\end{align*}
