Marsden and Hughes give their definition of an integral curve as such:
(i) Let $U \xrightarrow{\quad w \quad} T_S $ be a vector field, where $U$ is an open subset of $S$. A curve $g \xrightarrow{\quad c\quad} S$, where $g$ is an open interval (1), is called an integral curve of $w$ if for every $r$ in $g$
\begin{align} {dc \over dt} (r) = w(c(r)) \end{align}
I mostly understand this definition, but I'm a little uncomfortable with the definition of $g$. Specifically, by "an open interval", is this any open interval that is diffeomorphic to the unit interval $(0, 1)$. That is, $g \simeq (0, 1)$, and this open interval has nothing to do directly with the open set $U$.
Curves are usually defined so that their domains are intervals in $\mathbb{R}$, so I'd guess by "open interval" they mean "open interval in $\mathbb{R}$", i.e. $(a, b)$ for some $a < b$. (Looking at the source, they do mention earlier on p. 94 that "$g$" (which is actually a script $I$, e.g. a fancier version of $\mathcal{I}$) is "an open interval of the real line".
(However, if the domain of $w$ is $U$ and not all of $S$, I'd expect that they mean for the codomain of $c$ to lie in $U$ and not $S$, otherwise the expression $w(c(r))$ doesn't make much sense.)