Section 9.4

Arc Length in Parametric

If we think of as time, the length of the path is the integral of the speed. Speed is determined by the Pythagorean theorem of the velocity vector components.

The Arc Length Formula

The Key Formula

Arc length = ∫ speed dt

Why It Works

If is time, then and are velocity components. The Pythagorean theorem gives us speed.

Speed Formula

✨ Cleaner Than Cartesian!

Compare to the Cartesian formula . The parametric version treats and symmetrically — no confusing fractions!

Worked Example

Circle Circumference

1. Derivatives

2. Speed

3. Integral

Level Up Examples

Length of One Arch of a Cycloid

Step 1: Find the Derivatives

Step 2: Compute the Speed²

Step 3: Use Half-Angle Identity

Using :

(since sin(θ/2) ≥ 0 for 0 ≤ θ ≤ 2π)

Step 4: Integrate

Let , so :

Final Answer

The arc length is exactly 8 times the radius of the rolling circle!

Speed of a Particle

A particle's position at time is given by

Find the speed at any time t

Find the speed at point (5, 4)

The particle is at (5, 4) when and :

From →

Speed at :

Speed at (5, 4)

(if distance is in meters and time in seconds)