Without Green’s theorem, why is $\oint_C(xdy) = \Delta A$

Question

A textbook that I am reading states that $\oint_C(xdy) = \Delta A$ and I do not understand why. Any other place on the internet uses Green’s theorem, but the book is using this identity to derive the theorem, so they end up being “useless”. Can anyone help?

Answer 1

WLOG imagine a closed loop that can be neatly split into a "right" and "left" curve (for exotic shapes that cannot, split the curve into regions that do have a neat "right" and "left" boundaries i.e. a Type II region). The line integral becomes

$$\oint_C x(y)\:dy = \int_a^b x_R(y)\:dy + \int_b^a x_L(y)\:dy = \int_a^b x_R(y) - x_L(y)\:dy$$

where $a<b$ are the upper and lower extremes of a Type II region. Then we recognize that this integral is the end result of a double integral on a Type II region.

$$ = \int_{y=a}^{y=b} \int_{x=x_L(y)}^{x=x_R(y)}1\:dx dy = A$$

And since any path connected open domain can be split into some number of Type II regions, this method will always work. The line integrals on the borders of the neighboring regions will be of opposite orientation and thus cancel out, leaving only the line integrals on the original boundary uncanceled.