Area

Recall that the integral is the area between the area between a curve and the x axis.

To find the area of , we divide the interval into tiny angular increments of .
With each small interval , , we pick a value at which to evaluate .
We use that value to define the radius of a sector of a circle.
Recall that the area of a sector of a circle is .
We know that the sector has area , where represents a differential.
We can approximate the area "inside" the curve as the sum of the areas of a multitude of sectors of different radii, and setting gives us the exact value.

Definition

The area of a polar curve between and is:

and the area between to curves and with on is

Example

Find the area of the portion of the circle which lies above the line .

Solution we will use polar coordinates since it's somehow easier.
The circle has polar equation , and the line we have .
We also need to know the range of value of . For this, we can just observe that the curves intersect when . Therefore, runs from to .