PX275 - B2a - 1D integrals in 2D and 3D

consider a path in the $x - y$ plane
the position can be written as a function of $s : x (s); y (s)$ , such that $d s^{2} = d x^{2} + d y^{2}$
more useful to write:

\begin{matrix} d y = \frac{d y}{d x} d x \\ d s^{2} = d x^{2} + {(\frac{d y}{d x} d x)}^{2} \\ d s = d x \sqrt{1 + {(\frac{d y}{d x})}^{2}} \end{matrix}