Is there simple layman's definition of integration that doesn't refer to a graph?

Factual Questions
61

I think what’s going on here is that we, in our ordinary lives, spend a lot of time talking about quantities measured in distance per time and not a lot of time talking about quantities measured in distance times time. So it’s common to talk about speed (miles per hour) but it’s not common to talk about absement (mile hours).

And because of this, and thinking of functions as often something like distances at times, some people are much more comfortable with thinking intuitively about rates of change of f(x) with respect to x directly (“Why, that’s just a speed! Miles per hour!”), but not with thinking about accumulations of f(x) with respect to x directly (“That’s a… what’s that? ‘Mile hours’? Huh?”).

But you can do it. You can do it fine. There’s no need to bring graphs into it. The mathematics is what it is, and could be perfectly well understood by an entirely non-visual, non-spatial alien mathematician.

62

All of this stuff is just the limit of discrete analogues.

The discrete analogue of differentiation is (f(x) - f(y))/(x - y), the average rate of change of f as its input changes from x to y (“rate” meaning we divide the change in output by the change in its input). Continuous differentiation is the limiting value as x and y become infinitesimally close.

The discrete analogue of integration is f(x) * (p1 - x) + f(p1) * (p2 - p1) + … + f(y) * (y - pn), the weighted total of f as its input changes from x to y, stopping at points p1, …, pn along the way (the weighting being such that each amount of output is multiplied by the amount of input comes from). Continuous integration is the limiting value as we take infinitely many points covering everything between x and y.

In the simplest discrete case, we use step sizes of only 1, so that differentiation becomes f(x + 1) - f(x) (the difference between two adjacent values of f) and integration becomes f(x) + f(x + 1) + f(x + 2) + … + f(y) (the sum of many adjacent values of f). But the step sizes become infinitesimally small as we move towards the continuous limit, and this introduces the complication of dividing or multiplying by the step sizes (in differentiation or integration, respectively) to maintain a roughly stable value even as the step sizes change. Still, conceptually, we are looking at subtractions vs. additions, just “weighted” such.

63

For what it’s worth, my primary mental image for that sum is a pie chart.

And this is the first I’ve ever heard of absement. Could you give one or two real-world examples where it comes up?

64

I’d never heard the word before either, but the first thing that came to mind (and, expectedly, mentioned in the Wiki article), were PID controllers. PID is “proportional-integral-derivative”, which are the three inputs used to compute the control output. The integral part could easily be in units of “meter-seconds” and reflects the average distance spent away from the target value. If this number gets too large, it means you have a consistent bias that needs to be corrected for.

65

To explain a bit more about PID controllers:

Suppose we have a boat on a (1-dimensional) river and we want to build a system that keeps the boat at a particular point. The boat just has a throttle (labeled in units of m/s) that will move it forward and back (it works just as well in reverse as forward).

A simple system is a “P” controller, which is to say we make the throttle proportional to the amount of error. Say we’re 100 meters downstream of the target and our constant is 1 s^-1. So we set the throttle to 100 m/s, though it probably doesn’t go that fast. Instead, we cruise at 10 m/s until we get close. At 10 meters, we start reducing the throttle. At 1 meter, we’re only going 1 m/s, and so on. It will take a while but eventually we’ll get as close as we want to the target. If we overshoot, then our positive constant means we reverse the throttle and back up.

What if the river is moving at 1 m/s? Well, we’ll still get close. But if you work it out, you can see that the stable point is 1 meter short of the target. At this point, the throttle is also at 1 m/s and we don’t move relative to the target because the river is going the same speed. Everything is stable but there’s a persistent 1-meter bias.

The answer is to add an “I” term. This is in units of meter-seconds. We integrate the distance error over time and add that (times a constant) to our throttle. It doesn’t have to be a large value; say 0.01 s^-2. Over 10 seconds, our 1 meter error gives 10 meter-seconds, which times our constant of 0.01 gives 0.1 m/s. So we’ve given the throttle a little nudge forward which inches us to the destination.

Given enough time, even tiny errors in position will build up (it’s an integral after all), so we can hold position arbitrarily close to the target regardless of the speed of the river.

The “D” term is also useful but mainly for improving dynamic response time. If there’s a sudden change in river speed, say, or our target moves, then the D term gives a boost.

66

Interesting-- I’ve encountered that basic concept before, but wasn’t aware that it was something standard and named. When I was in grad school, some of the undergrads, for their final project in an electronics class, built an electromagnetic levitation system (you put a permanent magnet under the electromagnetic, let it go, and it adjusted the electromagnet to hold it in place). But unlike the ones that you see for sale, theirs actually worked, and didn’t collapse one way or the other the moment you looked at it funny (and as an extra bonus, didn’t need a photocell to sense the suspended object, but sensed it using the electromagnet itself). I don’t remember all of the details about how they did it, but it was probably something like the PID systems you’re describing.

67

That’s likely. I built PID systems before having any idea of the name or the math behind them. The fundamental concepts are simple enough that you’ll run across them in basically any kind of control system. Proving that the system is stable, putting bounds on the response time, and so on requires some formalization, but putting together an ad-hoc system that uses PID (or some subset) as input isn’t terribly difficult (making it work well [particularly in a dynamic system] is a challenge, though, so props to the students).

The output of a PID controller can be anything, of course: position, velocity, acceleration, angle, temperature, mass flow, gas concentration, pressure, and so on. So the integral for just about any physical unit with respect to time is sensible here.

The Wiki page on absement provides an amusing list of additional integrals:
Absounce
Abserk
Abseleration
Absity
Absement
Displacement
Velocity
Acceleration
Jerk
Jounce

I am having a hard time thinking of uses for absity and above…

68

A thought I’ve often had, which seems relevant but in a way I’m not entirely sure how to draw out, is that there are three steps involved in differentiation as it’s normally conceived, and correspondingly three steps involved in integration, and it’s useful to see these separated and how they operate:

By a punctile function, I’ll mean any old function whose inputs are individual points, whatever that means. By a sum-extensive function, I’ll mean a function whose inputs are regions, and such that whenever smaller regions are combined to make a big region, the function’s value on the big region is the sum of its values on the smaller regions. By a mean-extensive function, I’ll mean a function whose inputs are regions, and such that whenever smaller regions are combined to make a big region, the function’s value on the big region is the weighted average of its values on the smaller regions, each region weighted in proportion to its size.

The first step in differentiation is this: you have a punctile function. You turn this into a sum-extensive function defined over intervals, which gives the difference in the function between the endpoints of the interval.

Another step in differentiation is this: you have a sum-extensive function. You turn this into a mean-extensive function, simply by dividing its outputs by its inputs’ sizes.

The last step in differentiation is this: You have a mean-extensive function. You turn this into a punctile function, whose value at a point is the limiting value of the mean-extensive function as its input regions shrink towards that point.

(We can do all this in various higher-dimensional ways, not just for intervals, but I’ll talk in terms of intervals for now)

So differentiation goes punctile original -> sum-extensive differences -> mean-extensive rates -> punctile instantaneous rates.

But many physical examples involve seeing none of the functions involved directly except the sum-extensive differences one.

When we measure something’s speed, we typically have direct access to the distance it has travelled over particular time intervals, but nothing else. (Rulers and metronomes feel primitive in ways which speedometers don’t). We then infer by calculation its average speed over those time intervals (dividing by the amount of time taken), or its instantaneous speed at particular times (approximated by using very small time intervals). And we do not even start with a punctile original here (we do not have some distinguished 0 point in the world; we measure distances as differences from the start).

Integration is the opposite of this:

The first step in integration is to take a punctile function, and turn it into the mean-extensive function giving its average value over intervals.

The next step in integration is take a mean-extensive function, and turn it into a sum-extensive function, simply by multiplying its outputs by its inputs’ sizes.

The last step in integration is to take a sum-extensive function defined over intervals, and turn it into a punctile function, by choosing an arbitrary point and function value at that point (the famed +C), and then assigning values at other points in accordance with the specified differences. [This last step can also fail to be possible in higher-dimensional contexts, the difference between “conservative” and “non-conservative” fields…]

This last step of integration as ordinarily conceived is least important, since it involves those arbitrary choices [and can fail to be possible]. So let’s ignore it. Integration, thus, goes punctile original -> mean-extensive averages -> sum-extensive accumulations.

And in the same way, let us ignore the first step of differentiation, so that we conceive of it as taking sum-extensive differences -> mean-extensive rates -> punctile instantaneous rates.

As noted before, I think the thing we witness most directly in many familiar physical cases is the sum-extensive quantity, and the others are inferred from this.

So differentiation takes a thing we witness directly, and produces from it the corresponding calculated entity. While integration conversely takes a calculated entity, and produces from this the corresponding sort of thing we would witness directly. And somehow, this seems less natural. The thing we would witness directly we would start by witnessing, not end by seeking to find out. And so this maybe has something to do with why differentiation seems conceptually natural and integration conceptually unnatural for some?

My thoughts are inchoate, I don’t know, but something like this may be in play. This is not a polished post, this is a stream of thought ramble, but I’ll put it out there for now.

69

I don’t mean that differentiation IS more natural than integration, mathematically or conceptually; only that it may be felt more natural from the point of view of certain kinds of physical illustrations people like to refer to.

Any example of a physical differentiation from quantity A to quantity B could just as well be thought of as a physical integration from quantity B to quantity A; the examples of physical differentiation and of physical integration are in precise correspondence. But when A and B differ in how much they feel directly witnessed, tangible vs. calculated, etc., there may be a preferred direction for thinking of these examples; hence people taking the relation between displacement and speed to well illustrate the meaning of differentiation and not the meaning of integration. And perhaps the bias is pervasively towards reading each such physical example as a differentiation instead of an integration.

70

Personally, I view integration as more natural.

Integration as accumulation has many physical analogues. The immediate visual is a barrel of water which has an input and output flow. This analogy maps to many physical systems.

Differentiation of course has the common distance->speed example. In simple cases this is also intuitive. But as soon as things start changing, it gets a little weird. We have to deal with the “paradox” of “how can you have a speed over zero distance?” There are solutions, but they’re a bit subtle and require either (seemingly) non-physical objects like infinitesimals, or techniques (limits) which are not necessarily comfortable to the non-mathy. Integration doesn’t have that problem to the same extent; there are many non-calculus techniques to find areas of common shapes.

Derivatives have problems with discontinuous functions. While this can lead to physical insights (water hammer, inductive spikes), the fact that the derivative may not be defined for an otherwise legitimate input is… uncomfortable.

Conversely, integration is always well-behaved for reasonably physical inputs. We can shut our flow on and off instantly with no problem.

That’s just me, of course. As Indistinguishable notes, everyone has prebuilt biases that affect their view on things. A lot of it is just exposure.

71

This went by unremarked as the mathematicians were picking up speed (dv/dt > 0 :))

Really it’s the simplest baseline explanation. Thank you for such a concise and distilled answer.

Derivative = rate of change of some whatever.
Integral = cumulative effect of the same whatever.

Given a difference in cumulative effect over an interval we can trivially compute an average rate of change.

Given an average rate of change over an interval we can trivially compute the cumulative effect.
That’s it. We’re done.

At least conceptually. Starting from here one can drive off into all sorts of useful abstractions. But this demonstrates clearly how anyone who’s arguing that somehow derivation is a simpler, more primary, operation than integration is simply missing the bus.