Short on time? This page is the recap. For the derivation, the diagram and worked examples, use the full study guide. BC-only material. Prerequisites (parametric derivatives 9.1, arc length of y = f(x) 8.13, definite integrals, substitution 6.9) are on the Calculus BC roadmap.
Recap
- A tiny piece of curve is almost a straight line with legs Δx ≈ x′(t)Δt and Δy ≈ y′(t)Δt.
- Pythagoras gives Δs ≈ √(x′(t)² + y′(t)²) Δt. Adding the pieces gives the integral.
- The limits are values of t.
- Exact answers happen only when (dx/dt)² + (dy/dt)² is a perfect square. Otherwise use the calculator.
Key relationships
| Situation | Formula or fact | Note |
|---|---|---|
| Parametric curve, a ≤ t ≤ b | L = ∫ from a to b of √((dx/dt)² + (dy/dt)²) dt | Smooth, traced once |
| Curve y = f(x) | L = ∫ √(1 + (f′(x))²) dx | Special case x = t (Topic 8.13) |
| Circle x = r cos t, y = r sin t | Integrand = r; 0 ≤ t ≤ 2π gives 2πr | Useful check |
| Traced twice | Integral = twice the length | Integral counts every pass |
| Any curve | L ≥ straight-line distance between end points | Quick sanity check |
Assumptions
- dx/dt and dy/dt are continuous on [a, b].
- The point does not go back over any part of the curve. An always-increasing x(t) or y(t) proves this.
- Radians for any trigonometric component.
Mistakes to avoid
- No square root, or adding dx/dt + dy/dt without squaring.
- Limits in x or y instead of t.
- Lost chain-rule factor: d/dt[sin(3t)] = 3 cos(3t).
- Splitting a root: √(a² + b²) ≠ a + b.
- Retracing ignored: a circle traced twice is not twice as long.
- Degree mode on the calculator.
Quick self-check
- Write the length integral for x = t³, y = e^(2t), 0 ≤ t ≤ 1. (∫ from 0 to 1 of √(9t⁴ + 4e^(4t)) dt)
- Find the length of x = 3 cos t, y = 3 sin t for 0 ≤ t ≤ π/2. (Integrand 3, so L = 3π/2, about 4.712)
- For x = ½t², y = ⅓(2t + 1)^(3/2), what is (dx/dt)² + (dy/dt)²? ((t + 1)², so the integrand is t + 1)
Next: practice questions.