2.1. Compute the integral

$$
\int_{A B} \frac{d l}{\sqrt{|x|+|y|}}
$$

along the segment connecting points $A(0,-2)$ and $B(4,0)$.
Solution. The segment connecting points $A$ and $B$ can be represented by the parametric equations $x=4 t, y=-2+2 t, t \in[0,1]$. Therefore,

$$
d l=\sqrt{d x^{2}+d y^{2}}=\sqrt{16+4} d t=2 \sqrt{5} d t
$$

and the desired line integral reduces to a definite integral:

$$
\begin{gathered}
\int_{A B} \frac{d l}{\sqrt{|x|+|y|}}=\int_{0}^{1} \frac{d t}{\sqrt{4 t+2-2 t}}= \\
=2 \sqrt{\frac{5}{2}} \int_{0}^{1} \frac{d t}{\sqrt{t+1}}=\left.\sqrt{10} \cdot 2 \sqrt{t+1}\right|_{0} ^{1}=2(\sqrt{20}-\sqrt{10}) .
\end{gathered}
$$