TITLE: Intuition behind arc length formula
QUESTION [11 upvotes]: I understand the arc length formula is derived from adding the distances between a series of points on the curve, and using the mean value theorem to get:
$ L = \int_a^b \sqrt{ 1 + (f'(x))^2 } dx $
But is there an intuition here I'm missing?  Something about taking the integral of the derivative seems like it should mean something..

REPLY [15 votes]: It might help instead to think of our function as a parametrized curve in the plane.  In other words, consider the curve $(x(t), y(t))$ for $t\in[a,b]$.  Then it's clear that the differential $ds$ which lies along the curve is given by $ds=\sqrt{dx^2+dy^2}$.
Now we integrate $ds$ with respect to $t$ and the integral becomes $\displaystyle \int_a^b{\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}}dt$.
Your formula follows by setting $x(t)=t$, and $y(t)=f(t)$.