The line integral is given by
By Green's theorem, the line integral along
is equivalent to the double integral over
(the region bounded by
)
Now consider the function
. We can think of the double integral above as a volume integral; namely, it's the volume of the region below
and above the region
in the
-
plane (i.e.
). This volume will be maximized if
is taken to be the intersection of
with the plane, which means
is the ellipse
.
For the double integral, we can convert to an augmented system of polar coordinates using
where
and
. We have the Jacobian determinant
So the double integral, upon converting to our polar coordinates, is equivalent to