which is much easier to work with. The original formula comes from the Pythagorean Theorem, actually. The idea is that if you zoom in really close on any curve, a tiny part of it will look like a straight line, whose distance is given by .
where the small change in is and the small change in is . To find the total length of the curve, you just add up all those little parts:
Letting changes this into an integral (recall Riemann sums).