I'll talk about how to integrate a scalar function $f(x,y,z)$ along a curve in $\mathbb{R}^3$ using the scalar arc length element $ds=| \, d\vec{r} \, |$. The trick (14.2) is to make a change of variables using the parametric equations of the curve: $$\int f(x,y,z) \, ds = \int f(t) \, | \, \vec{r}^{\, \prime}(t) \, | \, dt. $$ This type of integral can be used to compute physical properties of wires (think length, mass, and center of mass).