Answer
$$\frac{5\left(13\sqrt{65}-\sqrt{5}\right)}{2}$$
Work Step by Step
\begin{aligned}
s &=\int_{a}^{b} \sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}} d t \\
s &=\int_{1}^{4} \sqrt{36 t^{2}+144 t^{4}} d t \\
&=\int_{1}^{4} \sqrt{36 t^{2}\left(1+4 t^{2}\right)} d t \\
&=\frac{6}{8} \int_{1}^{4} 8 t\left(1+4 t^{2}\right)^{1 / 2} d t\\
&= \frac{3}{4}\frac{2}{3}(1+4t^2)^{3/2}\bigg|_{1}^{4}\\
&= \frac{5\left(13\sqrt{65}-\sqrt{5}\right)}{2}
\end{aligned}
