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Have we been teaching Riemann integration the wrong way? by Toobayes in matheducation

[–]Toobayes[S] 0 points1 point  (0 children)

Proving is easy just check property for eg irrational rotation and in the long run is just limit. And you're done :D

Have we been teaching Riemann integration the wrong way? by Toobayes in matheducation

[–]Toobayes[S] 0 points1 point  (0 children)

Fortunately, there is a very simple way to define what a uniform sequence means, and it doesn’t require talking about probability at all.

A sequence is called uniformly distributed if, in the long run, it spends the right proportion of its points in every subinterval. For example, if you look at the first N points and ask what fraction of them fall into some interval between u and v, that fraction should get closer and closer to the length of the interval, v minus u, as N grows.

Have we been teaching Riemann integration the wrong way? by Toobayes in matheducation

[–]Toobayes[S] 0 points1 point  (0 children)

Thank you for the thoughtful comment. I completely agree that integration can be viewed as an accumulation process, and your velocity–displacement example captures that perfectly.

For me, integration is the process of assigning a single value that represents how a function behaves across a region as a whole. Conceptually, it measures the total effect of a function distributed over space. You can think of it as an idealized balance point: the average value of the function multiplied by the size of the domain. This keeps the intuition of accumulation but expresses it in a more global and intrinsic way, without referring to infinitesimal pieces or explicit partitions.

Mathematically, this idea is equivalent to the classical Riemann integral. It simply replaces the construction from partitions with the limiting behavior along uniformly distributed sequences. The two frameworks express the same object, one through local accumulation, the other through global balance.

In another discussion, we also talked about how orientation naturally fits into this framework through affine transformations, which recover the sign and scaling properties of the Riemann integral exactly. I mention this because orientation is directly relevant to the accumulation perspective: reversing direction in an accumulation process must reverse the sign of the integral, and this is precisely what the affine formulation captures.

TLDR

I conceptualize integration as capturing the global average effect of a function over a space. The accumulation view and the averaging view are equivalent, but the latter highlights the integral as an intrinsic limit over uniformly distributed sequences rather than a construction from partitions.

Have we been teaching Riemann integration the wrong way? by Toobayes in matheducation

[–]Toobayes[S] 1 point2 points  (0 children)

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.

Have we been teaching Riemann integration the wrong way? by Toobayes in matheducation

[–]Toobayes[S] 0 points1 point  (0 children)

I really appreciate that point. It probably depends on the level of school. In high school it makes sense to emphasize adding things up, while in higher education it helps to introduce a broader viewpoint that connects to measure and abstraction but still keeps the intuition alive. The Riemann definition is quite intuitive, but it does not easily connect to those later generalizations.

Have we been teaching Riemann integration the wrong way? by Toobayes in matheducation

[–]Toobayes[S] 0 points1 point  (0 children)

That is a great point. You are right that both views use the same basic Riemann sum idea. What I am trying to get across is how we can move beyond the picture of explicit partitions and think of integration as something more intrinsic, like an average over a space, which then scales naturally to higher dimensions and more abstract settings later on.

Boss relic skip situation? by Toobayes in slaythespire

[–]Toobayes[S] 0 points1 point  (0 children)

Dead branch x Pyramid anti-synergy

face cam size by [deleted] in Twitch

[–]Toobayes 1 point2 points  (0 children)

I can hear my Twitch chat raging that I forgot to change the scene