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Daily Math Minute

Unit 8: Applications of Integration

Arc Length

Using a definite integral to find the arc length of a curve, unique to BC.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Area Between Curves & Volume

Measuring the Length of a Curve

The distance formula gives the length of a straight segment. A curve isn't straight — but zoomed in close enough, a tiny piece of it looks almost straight. Before reading on: could summing up infinitely many tiny straight-line pieces measure a curve's exact length?

Definition — Arc Length

A thin piece of curve between x and x+Δx is approximately a straight segment with horizontal leg Δx and vertical leg Δy ≈ f'(x)Δx. By the Pythagorean theorem, its length is √(Δx² + Δy²) = Δx√(1 + [f'(x)]²). Summing these and taking the limit as Δx → 0 gives the exact arc length: L = ∫ₐᵇ√(1 + [f'(x)]²) dx.
L=∫ab1+[f′(x)]2 dxL = \int_a^b \sqrt{1 + \big[f'(x)\big]^2}\, dx

Worked Example — Checking the Formula on a Straight Line

Find the arc length of y = 2x from x = 0 to x = 3, where the answer is already known from the distance formula. dy/dx = 2, so L = ∫₀³√(1+4)dx = 3√5 ≈ 6.708. The straight-line distance from (0,0) to (3,6) is √(9+36) = √45 = 3√5 — matching exactly, confirming the formula reduces correctly for a straight line.

Worked Example — The Arc Length of a Genuine Curve

Find the arc length of y = x^(3/2) from x = 0 to x = 4. dy/dx = (3/2)x^(1/2), so (dy/dx)² = (9/4)x, and 1+(dy/dx)² = 1+(9/4)x. L = ∫₀⁴√(1+(9/4)x)dx. Let u = 1+(9/4)x, du = (9/4)dx: when x=0, u=1; when x=4, u=10. L = (4/9)∫₁¹⁰√u du = (4/9)(2/3)[u^(3/2)]₁¹⁰ = (8/27)(10^(3/2)−1) = (8/27)(10√10−1) ≈ (8/27)(30.623) ≈ 9.073. Sanity check: the straight-line distance from (0,0) to (4,8) is √(16+64) = √80 ≈ 8.944 — smaller than the curve's arc length, as it must be, since the shortest path between two points is a straight line.

Tip

Check any arc length result against the straight-line distance between the same two endpoints — a curve's arc length can never be shorter than that straight-line distance.

Common Mistakes

  • Forgetting to square (dy/dx) before adding 1, using √(1+dy/dx) instead of √(1+(dy/dx)²).

    The formula comes from the Pythagorean theorem applied to Δx and Δy — the derivative has to be squared, matching how a leg length is squared before being summed.

Key Takeaways

  • Arc length sums infinitely many infinitesimal straight-line pieces, each derived from the Pythagorean theorem.
  • L = ∫ₐᵇ√(1+[f'(x)]²)dx reduces to the ordinary distance formula for a straight line.
  • A curve's arc length is always at least as long as the straight-line distance between its endpoints.

Summary

This closes Unit 8, and the AB-shared portion of this course. Unit 9 turns to functions defined by a parameter, vectors, and polar coordinates — representations a single y = f(x) equation can't always capture.

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