Complex analysis - area inside a simple closed curve

[sweetvirgogirl]

Let C be a simple closed curve. Show that the area enclosed by C is given by 1/2i * integral of conjugate of z over the curve C with respect to z.

the hint says: use polar coordinates

i can prove it for a circle, but i am not sure how to extend it to prove it for any given closed curve

 

[AKG]

How would you express the enclosed by an arbitrary closed curve as a double integral? Can you apply Green's theorem to turn this into a single integral over C?