Section 9.8

Area in Polar

In Cartesian, we integrate rectangles (). In Polar, we integrate circular sectors (), sweeping out area like a radar scan.

The Area Formula & The Visual

Summing Wedges, Not Rectangles

In Calc 1, we found area by summing vertical rectangles (). In Polar, we don't scan left-to-right. We scan rotationally, like a radar or a lighthouse beam.

Instead of rectangles, our basic building block is a thin circular sector (a wedge/slice of pizza). The area of a sector is . For a tiny angle change , the area is:

Key Steps

Sketch the curve (or use Desmos) to see the loop.
Find the bounds () where the slice starts and ends. Usually found by setting for loops.
Set up the integral using .

Visual: The graph shows a simple wedge being swept out.

Sweeping out a sector from to

Worked Example

Cardioid Area

Find the area of .

1. Bounds

The entire curve is traced from to .

2. Integral Setup

3. Result

Using , the integral yields .

Full area swept from to

Level Up Examples

A. Area of One Loop

Find the area of one petal of the Rose Curve .

1. Find Bounds (r=0)

.
.

2. Integrate

(Use Double Angle: )

Result:

B. Area Between Curves

Area inside and outside .

1. Find Intersection

.
.

2. Setup

Outer minus Inner:

Don't subtract radii first!
Subtract their squares.