Arc Length

Arc length is simply the length of a curved path: how far you'd go if you walked along the curve instead of cutting straight across. ("Arc" just means a piece of a curve; the word started out meaning a piece of a circle.) It's what you need to know how much road, cable or track a curved design takes. The formula $\int_a^b \sqrt{1 + [f'(x)]^2}\,dx$ looks intimidating, but it's just a few simple parts. $dx$ (a tiny step to the right), $f'(x)$ (how steep the curve is there, so $f'(x)\,dx$ is how far you climb on that step), the square root (Pythagoras, turning "over and up" into a straight-line distance), and the integral sign (add up all the tiny steps). That's the whole thing, so bear with the symbols.

You already know how to measure a straight segment: Pythagoras. A curve is harder, but this is the same game we played with area. There, we couldn't measure a curved region directly, so we cut it into rectangles we could measure and let the pieces get thin. Here, we cut the curve into short straight segments.

Put a polygon on the curve, with its corners sitting on the graph. A segment covering a horizontal step $\Delta x$ rises by about $\Delta y \approx f'(x)\,\Delta x$, since the slope is rise over run. So its length is

Add up the segments and let $\Delta x \to 0$, and that sum becomes the arc length integral. Notice what the formula is saying: where the curve is flat, $f'$ is $0$ and each piece is just $dx$ long. Where it's steep, the same horizontal step covers a lot more curve.

Curves written as $x = g(y)$. Swap the roles: $L = \int_c^d \sqrt{1 + [g'(y)]^2}\,dy$. Sometimes this version is much easier, so it's worth checking both.

The arc length function $s(x) = \int_a^x \sqrt{1 + [f'(t)]^2}\,dt$ measures distance along the curve from its start. By the Fundamental Theorem, $\dfrac{ds}{dx} = \sqrt{1 + [f'(x)]^2}$, which is just the speed at which you rack up length as $x$ moves.

A warning. That square root makes most arc length integrals impossible to do exactly, even for curves as simple as $y = x^2$ or $y = \sin x$. Textbook examples are picked so that $1 + [f'(x)]^2$ turns out to be a perfect square, and the root disappears. When it doesn't, use numerical integration.

Road and rail designers price materials by arc length. Animators and game engines move characters along paths at constant speed by re-parametrizing curves by arc length, which is exactly what the Bézier lab does.