where
is
the region whose area is to be found.
University Maths Notes: Calculus – 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 find
and
such
that
then
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.
Choose
and
so
thatthe
the right hand side becomes 1 and we have
We now transform to polar coordinates
and
By
taking
as
increasing from 0 to
we
are orienting the curve counterclockwise, hence in a positive
direction.
and
then
In practiceandare
chosen so that the final integration becomes tractable.
Example: Find the area of the triangle below.
Take
thenand
On
and
on
so
only the middle integral contributes to the area.
On BS,
so