Example 2. Calculate the length of the arc of the semicubical parabola $y^{2}=x^{3}$, intercepted by the line $x=5$.
Solution. The given arc consists of two parts symmetric with respect to the $O x$ axis (see Fig. 5.8).

Let's compute the length of one of them. By finding the derivative of the function $y^{2}=x^{3}$ and substituting it into formula (5.18), we get

$$
\begin{gathered}
\frac{1}{2} l=\int_{0}^{5} \sqrt{1+\left(\frac{3}{2} x^{\frac{1}{2}}\right)^{2}} d x=\int_{0}^{5} \sqrt{1+\frac{9}{4} x} d x= \\
=\frac{1}{2} \int_{0}^{5} \sqrt{4+9 x} d x=\frac{1}{2} \cdot \frac{1}{9} \int_{0}^{5}(4+9 x)^{\frac{1}{2}} d(4+9 x)= \\
=\left.\frac{1}{18} \frac{(4+9 x)^{\frac{3}{2}}}{\frac{3}{2}}\right|_{0} ^{5}=\left.\frac{1}{27}(4+9 x) \sqrt{4+9 x}\right|_{0} ^{5}=\frac{1}{27}(49 \cdot 7-4 \cdot 2)=\frac{335}{27} \\
l=\frac{670}{27}=24 \frac{22}{27}
\end{gathered}
$$