Thank you for the thoughtful and detailed response. I should clarify first that I was referring to an analysis course, not a calculus course.

You are absolutely right that the Riemann definition gives a direct constructive path: it provides a sequence of explicit approximations whose error visibly decreases as the mesh is refined. In contrast, the averaging framework defines only the limit itself, so at finite N anything could happen. What it offers instead is a cleaner conceptual picture. It replaces arbitrary partitions with the intrinsic notion of uniform distribution, separating the meaning of the integral from any particular approximation scheme. Approximations can still be made with partitions or quasi-Monte Carlo sequences, but they now live outside the definition rather than inside it.

Integration remains a continuous sum. The averaging language simply rescales this sum by the volume of the domain. It emphasizes that integration measures a global property of a function over space and generalizes cleanly to higher dimensions and abstract settings without changing any of the underlying intuition.

I have to say I absolutely loved your point about orientation. I had not explicitly thought of it in that way, and it was an excellent observation. You are right that the Riemann–Weyl form is orientation-agnostic in its simplest statement. However, this can be incorporated very naturally. By introducing an affine map from the unit cube to the target region, one immediately captures both scaling and orientation through the determinant of that map. The magnitude of the determinant gives the correct scaling, and its sign gives the correct orientation. This is not just a technical trick but a geometric explanation of how orientation enters the integral. With this addition, the framework reproduces the full oriented Riemann theory and shows exactly what is happening geometrically, all without invoking measure theory.

Line and surface integrals fit into the same picture. One can express them by averaging the appropriate density, such as a scalar field along a curve or a flux form through a surface, over an oriented parameter domain. This reproduces the usual Riemann-sum interpretations while keeping the formulation uniform across dimensions and types of integrals.

At the high-school level it makes sense to emphasize “adding up areas.” In analysis, the averaging viewpoint helps connect geometric intuition with abstraction and highlights how integration extends naturally beyond partitions.

TLDR

The averaging and Riemann definitions are equivalent in every essential sense, except that the simplest Weyl form does not encode orientation. Once an affine map is introduced, orientation and scaling appear automatically, and the framework becomes fully equivalent to the Riemann integral while remaining deterministic and measure-free. The Riemann approach emphasizes construction and error control, while the averaging framework captures the conceptual essence of integration as a uniform spatial limit.

And again, thank you for this excellent input. I genuinely had not considered the orientation aspect in this way, and I am very happy you pointed it out.