Section 8.1

Arc Length

Calculating the Arc Length is essentially the calculus version of the Distance Formula. Instead of finding the straight-line distance between two points, you are finding the "walking distance" along a curved path.

Imagine zooming in on a curve so far that it looks like a tiny straight line. By using the Pythagorean theorem on an infinite number of these tiny "straight" segments and adding them together, you get the exact length of the curve.

Previously, we could only find the length of straight lines. Now, with integration, we can measure any smooth curve!

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The Arc Length Formula

How it Works: The Geometry

For a tiny segment of a curve, the length can be thought of as the hypotenuse of a right triangle with sides and .

Using the Pythagorean theorem:

To turn this into a usable integral, we factor out a :

Arc Length Approximation

Sum of hypotenuses $\sqrt{dx^2 + dy^2}$

Segments: 4
As N increases, the red sum of lengths approaches the true black arc length.

The Formula

If you have a smooth function on the interval , the arc length is:

Key Requirements:

Differentiability

must be differentiable on the interval (no sharp corners).

Continuity

must be continuous (the "smoothness" requirement).

Step-by-Step Process

To solve an arc length problem, follow these steps:

1
Find the Derivative: Calculate .
2
Square the Derivative: Calculate .
3
Setup the Radical: Add 1 to your squared derivative: .
4
Set the Limits: Identify the start () and end () points.
5
Integrate: This is often the hardest part; arc length integrals frequently require -substitution, trig substitution, or algebraic tricks.
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Worked Example

Semi-Cubical Parabola

Find the length of on .

Step 1: Find the Derivative
Step 2: Square It and Add 1
Step 3: Setup the Integral
Step 4: Integrate

Use power rule with :

Answer:
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Level Up Examples

Example 2: Using a Trig Identity

Find the length of on .

Step 1: Find the Derivative
Step 2: Use the Pythagorean Identity!

This is a classic trig identity that simplifies beautifully!

Step 3: Integrate
Step 4: Evaluate
Answer:

Example 3: Integrating with Respect to y

Find the length of on .

Note: When the function is given as , use:

Step 1: Find
Step 2: The Algebra Trick!

Square it and add 1. Expand carefully:

This is a perfect square! The radical disappears!

Step 3: Integrate
Step 4: Evaluate
Answer:
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Practice Quiz

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