Poincaré invariant corresponds to area?

Question

In this paper(equation 34) click me it says that if we have a symplectic egg, then the area of a $(x_i,p_i)$ clone intersected with the egg is given by the poincaré invariant

$\int p dx = \int_0^{2 \pi} p_i(t) x_i'(t) dt,$ where $p_i,x_i : [0,2\pi] \rightarrow \mathbb{R}$ are a paramerization of the boundary of the area of intersection.

Unfortunately, I don't see how this number is related to the respective area.

Answer 1

There seems to be a sign discrepancy, but this is an application of Green's Theorem to give the area of a plane region bounded by a closed curve.