Example 1. Calculate the line integral of the first kind $\int_{L} \sqrt{x^{3} y} d l$, where $L-$ is the arc of the cubic parabola $y=x^{3}$, connecting the points $O(0,0)$ and $A(1,1)$.
Solution. We have $d l=\sqrt{1+y^{\prime 2}} d x=\sqrt{1+9 x^{4}} d x$. Therefore,

$$
\begin{aligned}
& \int_{L} \sqrt{x^{3} y} d l=\int_{0}^{1} \sqrt{x^{3} \cdot x^{3}} \cdot \sqrt{1+9 x^{4}} d x=\int_{0}^{1}\left(1+9 x^{4}\right)^{1 / 2} \frac{d\left(1+9 x^{4}\right)}{36}= \\
&=\left.\frac{1}{36} \cdot \frac{2}{3}\left(1+9 x^{4}\right)^{3 / 2}\right|_{0} ^{1}=\frac{1}{54}(10 \sqrt{10}-1)
\end{aligned}
$$

Answer. $\frac{1}{54}(10 \sqrt{10}-1)$.