Example 2.55. Calculate the length of the arc of the semicubical parabola $y=x^{3 / 2}$ from the point $x=0$ to the point $x=9$.
Solution. We will use formula (2.24).

$$
\begin{gathered}
L=\int_{0}^{9} \sqrt{1+\frac{9}{4}} x d x=\frac{4}{9} \int_{0}^{9} \sqrt{1+\frac{9}{4} x} d\left(1+\frac{4}{9} x\right)= \\
=\left.\frac{8}{9}\left(1+\frac{9}{4} x\right)^{3 / 2}\right|_{0} ^{9}=\frac{8}{9}\left(\frac{85}{4}-1\right)
\end{gathered}
$$