Section 9.9

Arc Length in Polar

To measure length along a spinning curve, we use the Pythagorean Theorem on the two components of polar motion: spinning () and moving out ().

1

The Pythagorean Derivation

Why is there an ?

Imagine walking along a polar curve. Your tiny step () is the hypotenuse of a right triangle made of two motions:

  • Radial Change (): Moving directly away from the center.
  • Tangential Change (): Moving sideways along the circle of radius .

By Pythagoras: .
If we factor out , we get the formula:

2

Worked Example

Cardioid Length

Find the perimeter of .

1. Setup Derivatives


2. Simplify


3. Result

Using identity :
Result = 8.

Computing the total perimeter length.

3

Level Up Examples

A. The Spiral

Length of from to .

1. Derivatives

.

2. Integrate


Factor out :

u-sub ()
Result:

B. The Circle Check

Verify has circumference .

1. Derivatives

.
.

2. Integrate

Circle is traced to via .

.
(Matches Geometry!)
5

Practice Quiz

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