Yes; this is why we say you cannot really integrate a function, but rather a density or a differential 1-form (nb after the integral sign one writes “f(x) dx”, not “f(x)”). Not sure if that phrasing clears anything up for the putative layman.
You could also talk about the relation between the integral and the “average value” of a function, but to be honest the “sum of a whole bunch of little pieces” intuition seems the most useful, as it works regardless of the nature of the “little pieces”; it could be a mass integral or a flux integral or whatever.
Ah but then there is an implied 2D graph is involved. The little pieces need to have area for that to be a correct statement. Saying “little pieces” is meaningless, they have to be little slices that have area or cubes that have volume.
Suppose you’re driving a car and you look at the speedometer as you’re gradually slowing down. The rate of change for your speed is a small negative number. But the car is still moving forward. The cumulative effect of your speed is that you continue to get further away from your starting point. The derivative of speed is acceleration, the integral of speed is distance traveled.
You can imply a graph if you want; I never did. The little pieces I’m referring to could just as well be pieces of mass, or of gravitational field, or of anything else that can be measured with numbers.
I don’t think this avoids the graph really. The idea of rate of change implies a “flow” of some kind from one value to another, and the most intuitive way to look at that is a graph.
Less woo-ey, differentiation and integration are machinations that work on functions, which are sets of ordered pairs (or which can be considered sets of ordered pairs). That input/output aspect is essential to the concept. Now, a graph is one way of organizing those ordered pairs, and so no matter what you do you can always find a way back to the graphs. No matter what anyone else says here, it will always be possible to make a fairly intuitive leap into the world of graphs, including the example you present as a counterexample.
The word “pieces” implies you are plotting something, conceptually. There is no such thing as a “piece” mass, that you can add up to calculate an integrate, that is meaningless. It only means something if you are plotting how mass changes with time (or some other variable) and then chopping up the resulting shape into pieces.
And, indeed, in the example I mentioned related to the probability density function, the integration means “the probability of climate sensitivity being between x[sub]1[/sub] and x[sub]2[/sub] is y%”. If the distribution is presented as a graph, it’s the area under the curve, but it’s a meaningful quantity that has nothing to do with any particular visual presentation.
Sure! The standard definition of a definite integral (as the limit of Riemann sums) doesn’t have to say anything about a graph.
Oh well.
You can think of “the integral from a to b of f(x) dx” as “For every value of x between a and b, multiply the value of f(x) by the infinitessimally small amount dx, and then add those all up together.” This doesn’t really make sense, let alone work as a rigorous definition, but I find that it’s a useful way of thinking about what a definite integral is doing.
OK, now I’m completely lost. No such thing as a piece of mass? Of course there is. In what sense is there not such a thing (and yet, in that same sense, there is a piece of area)?
Simple. A graph is a plotting of variables from a function. You can split the function and graph it, and then take the area of that portion of the function.
Nothing analogous exists for a mass. The only way you could do it is if you had a function that you turned into a mass. But then you would be graphing said function.
He asked for an answer without the graphing metaphor. He asked for an answer that stays within mathematics, the same way his definition of a derivative did.
Heck, your answer isn’t even accurate. Let’s say I cut a pie into a bunch of pieces, and keep trying to make them smaller, then put the pieces back together. Have I accomplished the mathematical concept of integration? No. I’ve performed the process of addition, or possibly infinite summation.
Maybe this will help. The last time I ever taught calculus before I retired I walked into class the first day and said that I would start with a couple of questions they could all answer:
If you drive at a steady velocity of 50 MPH for an hour, how far would you travel?
If you drive at a steady velocity for an hour and you travel 50 miles, what was your speed?
I then explained that what calculus was about was removing the word steady and replacing the speed/distance by variable functions. The first was integral calculus and the second differential.
And then, painting another coat, and another, until there’s two feet of paint on the board. How long would that take?
Or instead, imagine how thick the paint would be, after, I dunno, 10,000 coats.
Does anyone know how thick a coat of paint is?
I don’t, but I know it’s really thin - there’s a proverb - ‘Missed it by a coat of paint’.
It’s probably gonna vary, across the board. And based on things like temperature, the brush you use, and which part of the paint can you dip the brush from.
But imagine if you could precisely add up that (essentially) infinite number of all those infinitesimally small thicknesses.
That’s basically what an integral does (if you have an equation* that provides that precision).
Nitpick: no reason to restrict to analytic functions. You can even integrate functions that are not continuous.
Sure they are. (And I think that accurately describes the process: in order to find the weight of a pie, you cut it into a huge number of small pieces, weigh each bit, and add up all the individual weights.)
(Indefinite) Integration is the summation of one variable w.r.t another variable.
(Definite) Integrals are the sum of one variable over the range of another variable.
All you’re saying there is that anything analogous to a graph is a graph. But you don’t need a graph anywhere in this process. If you want to say that not having a graph makes it not analogous to a graph, OK, I guess, but who said it needed to be analogous to a graph?
Yes, you do have to have a function. And yes, given a function, you can make a graph of it if you’d like. But the same is true (on both counts) for derivatives, and the OP doesn’t seem to have a problem with them.
How is this not conceptually different from using a graph if defining differentiation as “the rate of change of one variable w.r.t. another variable” gets your “no graphs” approval?
I confess I don’t understand how to perform calculus, however Marcus de Sautoy gave a good simple explanation of its application in his highly recommended podcast series on BBC radio “a brief history of mathematics” (which I believe our foreign friends can download). It certainly made the concepts clear to me…to paraphrase…
An apple falls from a height x. It accelerates all the way through its fall. How can we find out its velocity at any given point during that fall? Well we can get an accurate* average *velocity by measuring the time for the whole fall and the distance it fell. To get a more accurate measure we can slice the fall into smaller sections and measure time and distance for the section that includes the point we are interested in. To be even more accurate we create smaller and smaller slices until we create an *infinitesimally * small slice that relates to the point we are interested in. That gives us the velocity of the apple at that point.
Now in the framework of that example, to what do the terms “integral” “integration” and “differentiation” refer?
Oh well.