Arc Length — Mathematics Knowledge System

Overview

The arc length formula computes the exact length of a curve y=f(x) over an interval [a,b] by integrating the local stretching factor √(1+[f'(x)]²), which accounts for how much longer a small piece of curve is than its horizontal projection.

Interactive Graph

Formal Definition

Definition

For a smooth curve y=f(x) on [a,b], the arc length is:

L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx

Arc Length Formula

Worked Examples

Compute f'(x) = (3/2)x^(1/2), so [f'(x)]² = (9/4)x.
$f'(x)=\frac32 x^{1/2},\quad [f'(x)]^2 = \frac94 x$
Set up the arc length integral.
$L = \int_0^4 \sqrt{1+\frac94 x}\,dx$
Substitute u=1+(9/4)x, du=(9/4)dx.
$L = \frac{4}{9}\int_1^{10} \sqrt{u}\,du = \frac49\cdot\frac23 u^{3/2}\Big|_1^{10}$
Evaluate.
$= \frac{8}{27}(10^{3/2}-1)$

Answer: L = (8/27)(10√10 - 1) ≈ 9.07

Practice Problems

Difficulty 5/10

Set up (but do not evaluate) the arc length integral for y = x² from x=0 to x=2.

Difficulty 5/10

A highway horizontal curve is a circular arc of radius R = 300 m subtending a central angle of 40°. Find the length of the curve (the distance a car actually travels around it).

Difficulty 7/10

A cable hangs as a shallow parabola y = 0.01x² between towers at x = −50 m and x = 50 m. Set up the integral for its length, and explain why the cable is longer than the 100 m tower spacing.

Quiz

The length of a circular highway curve of radius R and central angle θ (in radians) is:

R + θR·θR/θR²θ

Why is the arc length of any sloped curve always greater than its horizontal span?

Because the integrand √(1+[f'(x)]²) is greater than 1 wherever the slope is nonzeroBecause integrals are always largeBecause of roundingIt isn't — they are equal

Summary

Arc length L = ∫ₐᵇ √(1+[f'(x)]²) dx accounts for the extra length a curve has compared to its horizontal projection.
The formula comes from approximating the curve with tiny straight-line segments (via the Pythagorean theorem) and taking the limit as the segments shrink — essentially a Riemann sum.
Many arc length integrals don't have elementary closed forms and require numerical methods or substitution techniques.

References

WebsiteWikipedia — Arc length