The curve arc length does not depend on the given parameterization of the this curve, because there are infinite ways to represent it according to the parameter t. For example
$$\vec{f}(t) = [t,t^2] \; ; \; \vec{g}(t) = [t^3,t^6]$$
they are both parametrizations of the same identical parabola. The arc length L, considering a regular curve in a range $[a, b]$, depends on a parameter $s$ that takes the name of curvilinear abscissa. Taking as an example a generic curve, and calculating the length of the arc between $0$ and $t$
$$s = s(t) = \pm \int_{0}^{t} ||r'(u)|| du$$
Forgive me the question, but in the texts I have never found anything exhaustive: why in the last formula, $||r||$ is written in function of another variable, in this case $u$?
Thank you in advance