Arc length function

The arc length function of a parametric equation is the function for which a given change of $\Delta t$ corresponds to a change in the arc length of the function by the same amount. It is defined as

$s(t)=\int\limits_0^t\|\vec{r'}(t)\|dt$

For example, say we have the vector parametric function

$\vec r(t)=\begin{bmatrix}2t\\\sin(t)\\\cos(t)\end{bmatrix}$

$\|\vec{r'}(t)\|=\sqrt{2^2+\cos^2(t)+\big(-\sin(t)\big)^2}=\sqrt{4+\cos^2(t)+\sin^2(t)}=\sqrt{4+1}=\sqrt5$

$s(t)=\int\limits_0^t\sqrt5dt=\sqrt5t$