= Solution
The <Hamiltonian> is $\boxed{H(q,p)=p^2/(2m)+V(q)}$. On a regular family of closed <phase-space trajectories>, <action-angle variables> are <canonical coordinates> $(\theta,I)$ in which $\theta$ is periodic modulo $2\pi$ and $H=H(I)$. <Hamilton's equations> then give $\dot I=0$ and $\dot\theta=dH/dI$.
The <action variable> is
$$
\boxed{I=\frac1{2\pi}\oint p\,dq=\frac1\pi\int_{q_-}^{q_+}\sqrt{2m(E-V(q))}\,dq.}
$$
The orientation is that of the motion, so the integral is the positive enclosed <phase-space area> divided by $2\pi$.
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