Is this a viable notion of integration? Does it have a name?

See postscript down below first.

I'll start with an illustrative example with regard to line integrals of vector fields. Suppose we have a vector field [;\mathbf{F};]and a curve [;C;]. The line integral of [;\mathbf{F};] over [;C;] is scalar valued and admits an intuitive description as being the sum of dot products of the form [;\mathbf{F}(\mathbf{r})\cdot d\mathbf{r};] where [;d\mathbf{r};] represents a small nudge along the curve and [;\mathbf{r};] a given position on the curve.

This appears to be highly suggestive of a possible object that can be more-or-less characterized as a vector valued analogue of the usual line integral: in particular, it could admit a representation such as [;\int_C{\mathbf{F}(\mathbf{r})}|d\mathbf{r}|;] and be characterized intuitively as the sum of values of [;\mathbf{F};] along the curve scaled by the arc lengths of the regions from which we pick representatives.

It is also worth considering whether or not this generalizes to higher dimensions (in retrospect it might have been better to contrast this with the surface integral).

Aside the questions in the title, provided the first has a positive answer, where does this notion see use?

P.S. Given that we can pretty much decompose the integral I'm proposing into a handful of independent scalar integrals depending on the dimensionality of our space, I've realized in the course of writing this that the notion probably is well-founded (and also not very interesting). It's still worth asking whether or not the idea has a name though.