A vector field is like a flow, but a 1-form $\omega$ is like a stack of planes (or lines in 2D). Misner, Thorne, and Wheeler describe 1-forms as "bongs of a bell" as you pierce through surfaces. The integral $\int_\gamma \omega$ counts how many of these lines you cross.
Integral $\int_\gamma \omega \approx$ 0.00
(Draw a path to count crossings)
Crossings: Moving "with" the form (up/right usually) counts as positive. Moving backwards counts as negative. Try drawing a closed loop for $d\theta$ around the center!