Math101learn.math101.ca

Calculus IIUniversity3 min read

Arc Length

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

Cheat sheet

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.

Conceptual meaning

A small displacement along a curve has horizontal component $dx$ and vertical component $dy$, so the Pythagorean theorem gives $ds=\sqrt{dx^2+dy^2}$. Dividing by the chosen parameter produces the speed factor inside the integral.

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

Graphical or geometric meaning

Polygonal segments joining nearby points approximate the curve. Their lengths sum, and in the limit their direction changes continuously. The integrand is always at least one in graph form, so arc length is at least horizontal span.

Common mistakes and why they fail

Verification and reasonableness checks

Compare with the straight-line distance between endpoints.
Confirm the speed integrand is nonnegative and has length per parameter units.
Check whether a substitution actually simplifies the radical.

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.

Practice

Find the length of $y=0$ on $[2,7]$.
Find the length of $y=mx$ on $[0,b]$, $b>0$.
What parametric quantity is integrated for arc length?

Answers and brief solutions

$5$.
$b\sqrt{1+m^2}$.
Speed $|\mathbf r'(t)|$.

Connections and next steps

Check your understanding

Try it yourself

Hints are part of learning. Open one whenever it makes the next step feel possible.

1 practice question

Question 1Compute graph arc length · Standard

What is the arc length of y=2x on 0≤x≤3?

End of lesson

Nice work making it this far.

Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.