While differentiation can be defined as the “the gradient of a graph at a point”, it can also defined in generic, abstract, terms without referring to a graph at all. i.e.:“it is the rate of change of one variable w.r.t. another variable”
But can integration be defined that way?
Is there any way to define integration without referring to something like “the area under a graph” (or saying “the inverse of differentiation”) or reverting to complicated mathematical definitions?
It’s the sum of a whole bunch of little pieces of something. Those little pieces could be little pieces of the region underneath some curve on a graph, but they could be little pieces of anything.
While that is true. Saying “Integration is the sum of a whole bunch of little pieces of something.” is hardly a succinct definition like: “differentiation is the rate of change of one variable w.r.t. another variable”. And it is hard to get from there to a succinct definition without starting to introduce graph terminology.
If you consider the function you have as a rate of change, the integral is a function that has that rate of change.
E.g. if you know your velocity then acceleration is the derivative, i.e. rate of change for that, and the integral is the distance traveled, i.e. it’s the thing that has a rate of change as described by your velocity.
Which is another way of saying integration is the inverse of differentiation.
That is the way most people visualize definite integrals, though. And it was succinct enough for Eudoxus and Archimedes. If you need a more rigorous definition, that might be a problem, as it will quickly evolve into a “mathematical definition” (I like Lebesgue’s integral).
I apologize if I misunderstood what you are really asking.
It isn’t just not a rigorous definition, it is not a definition, period. It is a simply a statement about integration. If you didn’t know what integration was, saying “Integration is the sum of a whole bunch of little pieces of something.” would not explain to you what it was.
Then no, there’s no simple definition.
The derivative is the rate of change of something, and the integral is something that has a rate of change.
Well they would have considered it heretical (possibly literally) to describe ANYTHING mathematical without referring to geometry of some kind. They were dead for many centuries before anyone described anything like abstract analytical integration.
OK. Well, you have to say what you are integrating, so let us say you are after the “integral of a function”. In that case, you could say the (definite) integral is the continuum limit of summing the values of the function.
Not a definition, you say? Granted, but your question is analogous to asking, in “generic, abstract” yet non-mathematical terms, how to define “summation”. If you can first do that, it might be clear how to define “integration” among similar lines.
That is kind of the conclusion I came to. I had to give a lecture on this, and ended up having to use “the area under a graph” terminology (which was not ideal as the type of domain I was describing could not be defined using a simple 2D graph)
It is just weird to me as it integration does define a “real world thing” in it’s own right, not just the reverse operation of differentiation.
Consider water flowing into a cylindrical bucket from a hose. The rate of change of height with respect to time is a measure of the instantaneous flow rate. The height itself is an integration or accumulation of the flow into the bucket.
So integration is the aggregate/accumulated value of a variable with respect to another variable.
I would be quite comfortable defining summation as “add up”, it might not satisfy Bertrand Russell but it would suffice for most people.
Then I want to say something like, to integrate a function you add up all the values of the function, but that is essentially exactly what Chronos already wrote in post #2.
In any example of integration, though, the “adding pieces up” interpretation should be clear, even if it is, and most cases are, not a simple instance of an “area under a graph”. The pieces could be anything, as sayeth Chronos. I emphasize that if the integral does not compute an area then there is no real reason to think of it as one.
Also, for one, me.
Just sayin’.
That would be a very simple layman’s definition of integration, but it would be wrong. That is the definition of summation not integration, the two terms are closely related but not the same any more than defining differentiation as one variable divided by another variable.
To me, this is the essential intuition you want. I would say it somewhat differently, though: Given the rates of change of a variable (with respect to another variable) on an interval, the integral computes the total change over that interval. (For integration on an interval, obviously. It would have to be adapted for more general integrals.)
I don’t know if this is particularly helpful, but if nothing else it’s a practical illustration of why the “area under a curve” interpretation is extremely pertinent. It has to do with the probability density function, which illustrates the probability (found on the y-axis) of some variable under examination having any given value (found on the x-axis). Now if you look down at the smaller graph, it shows a skewed distribution. One might be tempted to look at the peak of the graph (the mode) and declare that to be the “most likely” value, and while that’s the most likely single value, it’s pretty meaningless by itself, especially since the median in a positively skewed distribution is going to be much higher than the mode, and the variable is most likely to have a value towards the tail, higher than the mode.
The most meaningful way to assess a PDF like this is generally to take a range of values around the mean, and state the probability that the variable will be within this range. That probability is the cumulative or integrated probability of all values in the range, and it’s represented precisely by the area under that section of the curve.
One of the practical uses of the PDF is in climate science, to represent predicted probabilities of climate sensitivity. I guess this example sticks in my mind because there have been misinterpretations of what a PDF really means – which in particular overlook the importance of integrating over a range rather than a visual interpretation focused on the single point of the peak – which has led to some misrepresented claims, whether intentional or not.
That’s a pretty close. I would quibble “aggregate with respect to” is less meaningful than “rate of change with respect to”.