Usually textbooks show the formula in 2D and attention is given as to how the mean value theorem allows the introduction of a derivative into the equation.
Thus far I have not been able to find a formal proof of arc length in 3D.
I thought someone might have run into one on some occasion.
$\vec{r}(t)=<x(t),y(t),z(t)>$
$\frac{d\vec{r}}{dt}=<\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}> $
$d\vec{r}=<\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}> dt$
$|d\vec{r}|=\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2+(\frac{dz}{dt})^2} dt$
$L=\int \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2+(\frac{dz}{dt})^2} dt$