Definition:Arc Length

Definition
Let $y = f\left({x}\right)$ be a real function such that $D_xf$ is continuous on the closed interval $\left[a..b\right]$.

The arc length $s$ of $f$ between $a$ and $b$ is defined as:


 * $s:= \displaystyle \int_a^b \sqrt{1 + \left({\frac {\mathrm dy}{\mathrm dx}}\right)^2}\ \mathrm d x$

For an explanation of this definition and a proof that such an integral exists, see Continuously Differentiable Curve Has Finite Arc Length.

Intuition
The arc length of a curve can be thought of as how long the graph of the curve would be if you cut it at the points $\left(a,f\left({a}\right)\right)$ and $\left(b,f\left({b}\right)\right)$ and then straightened it out.