When we talk about the line integral
$$\int \overrightarrow v \cdot d \overrightarrow l $$
(where $\overrightarrow v$ is a vector function), the vector function is giving us tons of vectors. What's happening is that each vector is being multiplied by the infinitesimal displacement towards the next vector. Is it correct to say that the point that the vector is describing is being multiplied by the infinitesimal displacement? Or how do I picture this?
I'm very much so a visual learner, so I'm trying to imagine how this all works. I like the way we illustrate regular 2-D integration as the sum of infinitesimally thin rectangles. This is really good for a visual learner because we can think ''area of the base x height''.
When you are doing a line integral, your infinitesimals are no longer distances or lengths -- they are actually vectors.
You can think of the underlying mechanism here as being projection of your vector $\vec{v}(\vec{x})$ onto your infinitesimal displacement vector $d\vec{\ell}(\vec{x})$.
That's why these integrals are great for representing work done by a force field on a moving object: the field is applying some force in some direction. You figure out how much of this force is in the direction of movement at each point (you project force onto your infinitesimal displacement). Then, you "sum" all of these up (integrate).
Does that help?