I saw that the length of a continuously differentiable curve $\gamma$ in $\mathbb{R}^n$ with $\gamma(t) \neq 0$ is defined as $\int_a^b |\gamma^{'}(t)|dt$, as can be found here https://en.wikipedia.org/wiki/Curve#Lengths_of_curves .
I dont understand why does it really define length as we want it to be? I hoped someone could explain the idea behind the definition and give some intuition.