9.3 Finding Arc Lengths of Curves Given by Parametric Equations

Arc Length Formula for Parametric Curves

For a curve defined by x(t) and y(t) on interval [a, b], arc length is L = ∫ from a to b of √[(dx/dt)² + (dy/dt)²] dt. This comes from summing tiny straight-line distances between points as t varies, using the Pythagorean theorem on infinitesimal changes dx and dy.

Requirements & Setup

The curve must be smooth (dx/dt and dy/dt continuous, not both zero) and traced exactly once over [a, b] to avoid overcounting length. Always find dx/dt and dy/dt first, square them, add, then take the square root before integrating. This integral often requires a calculator for evaluation on the BC exam.

Connection to Speed

The integrand √[(dx/dt)² + (dy/dt)²] represents the speed of a particle moving along the path, since it's the magnitude of the velocity vector. So arc length is just the integral of speed over time — total distance traveled equals speed accumulated over the interval.

Key terms

Parametric arc length formula

L = ∫ from a to b of √[(dx/dt)² + (dy/dt)²] dt

Key requirement for the formula to work correctly

The curve must be smooth and traced exactly once over [a, b].

Physical meaning of the integrand

It represents the speed of a particle moving along the curve (magnitude of velocity vector).

Steps to set up the integral

1) Find dx/dt and dy/dt. 2) Square each. 3) Add them. 4) Take the square root. 5) Integrate over the interval.

Why square dx/dt and dy/dt before adding?

It comes from the Pythagorean theorem: distance² = (Δx)² + (Δy)², applied to infinitesimal segments of the curve.

Try an AP-style question

Which integral gives the arc length of a parametric curve for a ≤ t ≤ b?

1. ∫ₐᵇ √(x² + y²) dt
2. ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt
3. ∫ₐᵇ √(1 + (dy/dx)²) dt
4. ∫ₐᵇ ((dx/dt) + (dy/dt)) dt