We can accomplish this by breaking this flow through the curve into tiny bits of flow, then adding them all up to find the flow/flux through the entire curve. Letting the fluid flow for an infinitesimal change in time, we get a parallelogram.

From this we can use the normal vector to get only the perpendicular component of the vector field, the one flowing directly out of the curve at an instant.

It becomes clear that to apply this to the entire curve we now have to integrate, and we end up with the Flux Integral:
The Flux Integral gives the total flow of a vector field through a closed curve per unit time