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For one interpretation of the area inside a curve in phase space, see Arnold's Mathematical Methods of Classical Mechanics page 20. In case you do not have a copy of the book, he defines a function $S: (E_0 - \epsilon, E_0 + \epsilon) \to \mathbb{R}$ which gives the area inside the curve associated to an energy level $E$ (assuming this is well defined). The problem on page 20 asserts that $T = \frac{dS}{dE}$ where $T$ is the period of motion along the curve.