Arc Length

A rigorous, example-driven guide to arc length, including hypotheses, method choice, verification, and practice.

Arc length is approximated by short chords and refined toward the curve’s length. A curved path is overlaid with a chain of straight chord segments. A magnified chord has horizontal and vertical changes labelled to show the local Pythagorean length element.

The central idea

If $y=f(x)$ has continuous derivative on $[a,b]$, its graph length is $L=\int_a^b\sqrt{1+[f'(x)]^2}dx$. For $x=g(y)$ use $\int_c^d\sqrt{1+[g'(y)]^2}dy$. A smooth parametric curve has $L=\int_\alpha^\beta\sqrt{(x')^2+(y')^2}dt$ when it is traced once.

Definitions, hypotheses, and notation

The smoothness hypothesis rules out undefined velocities in the basic formula, though piecewise smooth curves can be handled by splitting. Arc length is independent of a regular reparametrization because a change of parameter changes speed and differential by reciprocal factors. It is not independent of repeated tracing: traversing the same arc twice doubles the integral.

Exact antiderivatives are uncommon for arbitrary curves. Before turning to numerical integration, look for a perfect square under the radical or a derivative that suggests substitution. Any numerical length should still obey endpoint-distance and coordinate-span lower bounds.

A dependable method and decision rule

Choose the variable or parameter that makes the derivative simplest.
Check smoothness and that the intended curve is traced once.
Compute the relevant derivative components.
Form the nonnegative speed $ds$ factor and set correct bounds.
Simplify inside the square root before deciding whether exact integration is possible.

Fully worked example

Common mistakes and why they fail

Length requires speed, not signed change

For a sufficiently smooth graph, $L=\int_a^b\sqrt{1+[f'(x)]^2}\,dx$ accumulates tiny tangent-segment lengths. The square ensures negative slopes still add positive length. A parametrized curve uses $L=\int_\alpha^\beta\|\mathbf r'(t)\|\,dt$ and counts repeated tracing as repeated distance. Simplify under the square root and look for a perfect square, but keep absolute values when taking roots. Length must be nonnegative and at least the straight-line distance between endpoints, giving two checks. Reversing a parameter changes velocity direction but not speed, so it cannot change length when the same path is traced once. Confirm the chosen interval does not unintentionally retrace part of the curve.