Let $(M,\omega)$ be a symplectic manifold. Let $f:M \to \mathbb R$ be a smooth function. We have vector fields $X_f$ defined by $\omega(X_f,)=df$. Let $\phi_t$ be the flow of $X_f$ and let $\gamma: \left[a,b\right] \to M$ be a smooth curve. We define $\psi(t,s)=\phi_t(\gamma(s))$. How do we compute $$\int_a^b \int_0^1 \psi^*\omega$$
I use the following expression for $ \psi^*\omega$, $$ \psi^*\omega=\omega(\partial_s \psi ds + \partial_t \psi dt,\partial_s \psi ds + \partial_t \psi dt) $$ I suppose $\partial_t \psi = X_f(\phi_t(\gamma(s)))$ and $\partial_s \psi = D\phi_t \circ (d\gamma(s)/ds) $ but I don't know what to do after. I am not allowed to assume that $\phi_t$ preserves $\omega$.