Distance Traveled Along a Curve

A straight line between two points is the shortest path, but a curve that winds back and forth covers more ground. A hiker's pedometer reads more than the straight-line distance across a valley because the trail climbs, dips, and bends — every wiggle adds length the crow-flies distance never counts.

To recover that true length, zoom in on a tiny piece of the curve. Over a short stretch, the curve looks almost straight, so that piece has length √(dx²+dy²), a tiny hypotenuse built from a tiny horizontal step and a tiny vertical step. Factor out dx: √(dx²+dy²) = √(1+(dy/dx)²) dx. Integrating this from one endpoint to the other adds up all those tiny hypotenuses into one total length. This is the same idea as integrating speed over time to get distance on a velocity graph: speed is how fast position changes per unit time, and dy/dx plays an analogous role, describing how steeply the curve rises per unit of horizontal travel.

This topic sits right after area between curves and volumes of solids, and it uses the same slicing logic: cut the curve into infinitesimal pieces, find a simple formula for one piece, and integrate. It also closes the loop with distance-traveled problems from motion, since both ask the same question — how much total path length accumulates — just with position given directly instead of through a velocity function.