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CalculusBy Educator Today

C37 · Unit C8 Applications of integration

Arc length and distance along a curve

BC extensionReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 8.13 · BC
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

For a smooth graph y equals f(x), small segment length is approximately square root of [( delta x) squared +( delta y) squared ], which leads to L equals ∫ₐᵇ square root of (1+[f minutes (x)] squared)dx. The integrand is nonnegative. A curve’s length differs from area under it, and many arc-length integrals require numerical evaluation. Specify the integral even when no elementary antiderivative exists.

Worked example

For y equals 3x+1 on [0,2], f minutes equals 3, so L equals ∫ sub 0 squared square root of 10 dx equals 2 square root of 10. The endpoints differ by (2,6), giving the same distance square root of 40.

Explore the mathematics

Arc length and distance along a curve

Build curve length from short Pythagorean segments.

Setting 3 of 5

Scroll sideways to see the full diagram.

Why is curve length different from area beneath a curve?xy1Marked input 0.5; displayed curve height 1.4142Arc length of y=x² on [0,1]

Predict before continuing

Why is curve length different from area beneath a curve?

Read the complete explanation
  1. Why is curve length different from area beneath a curve?
  2. A short segment has length √(Δx²+Δy²).
  3. Divide out Δx and take a limit: integrate √(1+(dy/dx)²).
  4. Here L=∫₀¹√(1+4x²)dx≈1.4789, greater than the straight-line distance √2.
See the idea: Arc length and distance along a curve
See the idea: Arc length and distance along a curveThe straight segment joins (0,1) to (2,7). Horizontal change 2 and vertical change 6 give length √(2²+6²)=2√10, matching ∫₀²√(1+3²)dx.xy000.52141.5628(0,1)(2,7)

The straight segment joins (0,1) to (2,7). Horizontal change 2 and vertical change 6 give length √(2²+6²)=2√10, matching ∫₀²√(1+3²)dx.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Do not integrate square root of (1+f squared); the derivative appears because segment slope determines local stretch.

Investigate and explain

Compare polygonal approximations with the arc-length integral for y equals x squared. Explain why straight chords underestimate the curved path.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.