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Using Green's Theorem to Find an Area

We can integrate to find an area using the expressionwhereis the region whose area is to be found.

Compare this with Green's theorem:

If we findandsuch thatthen the right hand side will be the expression for the area of

For example, use Green's theorem to find the area of the ellipse with cartesian equation

An ellipse is a simple (no holes) closed curve.

Chooseandso thatthe the right hand side becomes 1 and we have

We now transform to polar coordinatesandBy takingas increasing from 0 towe are orienting the curve counterclockwise, hence in a positive direction.

andthen

In practiceandare chosen so that the final integration becomes tractable.

Example: Find the area of the triangle below.

Takethenand

Onand onso only the middle integral contributes to the area.

On BS,so

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