#### Answer

$$\frac{22\sqrt{22}-13\sqrt{13}}{27}$$

#### Work Step by Step

Given $$y=x^{3/2}$$
The arc length given by
\begin{aligned} \int_{a}^{b} \sqrt{1+\left(y^{\prime}\right)^{2}} d x &=\int_{1}^{2} \sqrt{1+\left(\frac{3}{2} x^{1 / 2}\right)^{2}} d x \\ &=\int_{1}^{2} \sqrt{1+\frac{9}{4}} x d x \\
&=\frac{2}{3}\frac{4}{9}\left(1+\frac{9}{4}\right)^{3/2}\bigg|_{1}^{2}\\
&= \frac{22\sqrt{22}-13\sqrt{13}}{27}
\end{aligned}