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Help with parametric area formula for closed curve

I have read in a text that the area bounded by a closed curve can be given by S(x(t),y(t))=12∫0T(xy∗−yx∗)dt S(x(t),y(t)) = \frac{1}{2} \int_0^T {(xy^* - yx^*)dt} in parametric form.
I would like to translate this into polar coordinates, but as I cannot see how it is derived I cannot proceed. I have no problem with the standard area under a curve of the form S(x(t),y(t))=∫0Tyx∗dt S(x(t),y(t)) = \int_0^T {yx^*dt}, but for the closed curve, it seems that I am trying to calculate S(x,y)=12[∫0Yxdy−∫0Xydx] S(x,y) = \frac{1}{2}[ \int_0^Y {xdy} - \int_0^X {ydx}] which makes even less sense to me. What am I missing here?
Chris8888
...


A Course in Pure Mathematics by Maragaret M. Gow has a couple of pages on this covering both the parametric and polar forms. It's sections 21.8 & 21.9 in my volume.

It's too much to type out, and I don't have a scanner. You may get a copy in your uni. library, or failing that it's relatively cheap on Amazon.
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Chris8888
OP
Thank you so much. It is of course Green's Theorem. In polar coordinates this is not required unless there is more than one value of r for a given value of theta.

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