Section 12.9

Arc Length

Measuring the physical length of a curve traveling through space.

Introduction

To find the length of a curve in space, we integrate its speed. Think of it this way: Distance = Speed × Time. In calculus terms, we sum up infinitesimally small distances: .

Interactive: Trace and Measure

Dragging increases total length. For this helix, speed is constant .

Key Formulas

Arc Length Formula

The length from to is the integral of the magnitude of the derivative (speed).

Arc Length Function

Measures distance from starting point .

Reparameterization

Writing in terms of (distance) instead of (time).
Goal: Find and substitute back.

Worked Examples

Example 1: Length of a Helix (Level 1)

Find the length of from to .

Step 1: Derivative

Step 2: Magnitude (Speed)

Step 3: Integrate

Example 2: Arc Length Function (Level 2)

Find the arc length function for starting from .

This is a straight line, so we expect to be linear.

Result: (Distance is 5 times time).

Example 3: Reparameterization (Level 3)

Reparameterize (Circle radius 3) using arc length.

Find s(t):
.
.
.
Invert to find t(s):
.
Substitute back:
.
Meaning: If you walk units, you are at angle .