On page 348 of Shifrin's Multivariable Mathematics, for a 1-form $\omega=\sum F_{i}dx_{i}$ on $\mathbb{R}^{n}$ and a parameterized curve $C$ given by a function $\mathbf{g}:[a,b]\rightarrow\mathbb{R}^{n}$, he defines$$\intop_{C}\omega=\int_{[a,b]}\mathbf{g}^{*}\omega=\int_{a}^{b}F_{i}\left(\mathbf{g}\left(t\right)\right)g_{i}^{\prime}\left(t\right)dt.$$
Why does he write $$\int_{[a,b]}\mathbf{g}^{*}\omega$$ and not$$\int_{a}^{b}\mathbf{g}^{*}\omega\ ?$$
I vaguely recall the $\int_{[a,b]}$ notation as referring to an unsigned definite integral, but may have got that wrong.
Vaguely recall from where?! At this point, we've defined the integral of a differential form over a parametrized manifold (perhaps with boundary). The notation $\int_a^b f(t)\,dt$ is the classical notation for the Riemann integral of the function $f$.