In many cases this can be quite an intuitive definition.
The integral of “current” is “charge” (across a capacitor, say): the integral of “velocity” is “distance travelled”, the integral of “reaction rate” is “amount of the substance”, and so forth. In all these cases you have something accumulating or decreasing over time and the integral tells you about how much total you’ve accumulated.
Novelty Bubble, your example is a derivative. Velocity is the derivative of position with respect to time, which is to say that it’s the average velocity over some interval, in the limit where the interval gets very short.
In the case of a falling apple, the integral of the position with respect to time isn’t really something you’d do, because it doesn’t give any interesting or useful result. But if, instead of starting with the position, you had the velocity as a function of time (which is actually much simpler to calculate), then you could use the integral of the velocity to get the position. This is what people mean when they say that integration and differentiation are inverses of each other: Saying that v is the derivative of x is equivalent to saying that x is the integral of v.
Thanks Chronos, You have added to my knowledge.
Chronos is completely correct, but if you guys don’t find “integration is addition (of many tiny pieces, each suitably weighted)” pithy enough, perhaps you may like to think of the integral of a function over a region as its mean value over that region times the size of that region.
(For what it’s worth, thinking of integration primarily in terms of area under curves is terribly narrow and leads to lots of pointless confusions, in my experience of years and years teaching calculus at various levels.)
Ah, I see DPRK beat me to mentioning average values.
Anyway, integration is just what all the summists have been saying: carve the thing up into lots of little pieces, assign each piece its average value, then add these all up weighted by the size of the pieces. (Actually, the point of carving up into little pieces is only that, when the pieces are small enough, there’s little ambiguity in what their “average value” should be taken to be. Otherwise, like I said, we could just as well note from the start, integration is the average value of the whole thing times its size.)
And you can integrate lots and lots of things, not just things that have any natural area interpretation. Of course, you can play a word game where you call any kind of additive quantity some sort of generalized “area”, but there’s no need to do the gymnastics of thinking of integrations of, say, complex-valued quantities in areal terms, unless there is specifically something to be gained by doing so.
Some others:
Water flowing into a pool. Integral of flow rate is the amount of water that ends up in the pool.
Population growth. Integral of <birth rate - death rate> over time is the change in population over that time period.
Volume vs. area. You can calculate the volume of a shape by modeling the volume as a sum of thin layers. Imagine building up a sphere from many layers of spherical shells. Each layer is a spherical shell with radius r, which has a surface area of 4 pi r^2. Integrate that from r=0 to r=R and you get the volume of a sphere (3/4 pi R^3).
This post only matters to mathematician pedants and not laypeople searching for starting intuition for what familiar “integration” means:
For what it’s worth, when I say “times its size” above, I allow sizes to be taken from any kind of measure (measured in any kind of quantities), and “times” to be any kind of bilinear operation. Thus, we can consider line integrals of complex-valued functions, or whatever.
Mind you, the way I see it, any linear operator is a kind of integration operator, for some kinds of quantities with respect to some sort of measure. The most familiar integrations are of real-valued functions on some domain with respect to the essentially unique translation-invariant suitably normalized measure on such, but there are more things in the world.
These are all good examples too. griffin1977, you can also integrate with respect to things other than time. Anytime you have one variable changing along a gradient of another variable, you can use integration to find how much you have accumulated or decreased. For example, as you pull a spring for a longer distance, you have to exert more and more force: the total energy you put in the system is going to be the integral of force with respect to distance.
Now that’s pretty much all i know about integration but I’ve never seen it described so concisely.
As an aside, a friend of mine has a very old mechanical device for calculating the area of arbitrary shapes, just gears and ratchets. Tells me that there’s a way to do it by some kind of averaging of rates of change on the shape of the outline. Beats me how that works either.
It’s not strictly accurate, but an old math teacher of mine said something like:
“Differentiation is the rate of change, Integration is the amount of change”
That seems good enough for casual conversation, but it does seem that the number of words you have to add to get differentiation “correct” is smaller than the number you’d have to add for integration.
I wonder if that’s somehow because differentiation “happens” at a point, but integration happens over an interval (i.e. between one point and another).
There are different ways to construct one, but they all feature a little wheel that turns as you move a pointer around the boundary of your shape. It is indeed a mechanical integrator.
Yes, obviously the other variable doesn’t need to be time, but there needs to be another variable that you are integrating with respect to, which is why simple defining integration as “adding lots of little things up” is not correct.
Actually the lecture that prompted this question discusses (in passing) this awesome device:
Its a trolley that does numerical integration, it was used at Los Alamos before the ENIAC was was available.
The lots of things that you’re adding up are, themselves, the variable you’re integrating with respect to. The integral over the limits of a banana of d(banana) is a banana.
IMO there is an extra logical leap in the definition of integration, that doesn’t exist in the differentiation case. To define a “piece of something” you have to introduce the concept of area (or volume or whatever the higher-dimension equivalent is), which is implies you have a graph. Whereas that’s not the case with differentiation, it’s quite possible to imagine “the rate of change of one variable w.r.t. another variable” without a graph.
But why do you object to the analogous characterization “the total amount accumulated at a given variable rate (w. r. t. another variable)”?
It is quite possible to imagine “a piece of something”, accumulation, etc., without a graph. I do it all the time. (You often COULD graph things to illustrate it, if you like, but you could just as well graph things to illustrate rates-of-change if you like too. You don’t have to. It’s an unnecessary linkage.)
If I were to consider something like 1 + 1/2 + 1/4 + 1/8 + … = 1, am I necessarily visualizing a graph of an asymptotically descending staircase and the area bounded by it? I do not feel that way. I can think of the graph if I like, but am not forced to. And this is an integral. This is an integral on an infinite discrete space, but it is an integral. And integrals on continua operate no fundamentally differently. They are weighted additions, that is all. Sometimes you may wish to think of the values being added as heights, the weights as widths, and the calculation as producing an area. But often, there is no reason to think of it in such terms.
[FWIW, integrals over infinite regions have to be treated as limits of integrals over finite regions if one wants to make use of the mean-based perspective on integration I noted above, since division by infinite size is tricky. But this is fine.]
I don’t so much (it’s better than anything I could come up with without drawing a graph).
But I still don’t think I’d use it (if I repeat the lecture I’d still draw a graph) as I think the phrase “the amount one variable changes with respect to another” has more meaning than “the total amount accumulated at a given variable rate (w. r. t. another variable)”. To me it seems the former phrase doesn’t need further explanation, if you understand anything at all about functions it makes sense. The latter phrase needs to be expanded on somehow, what does “accumulated at a given variable rate” mean?
Do you have the same objection to talking about averages (in the sense of the mean, as I always clarify…)? Do you feel averages implicitly mean average height of a graph?
If I note that some function f(t) describes the temperature of my hometown over various times t, and then speak of “the average temperature of my hometown in the year 2016”, is the underlying meaning of that expression only possibly explained and grasped via “Well, if you were to make a graph, on which one axis was time and the other axis was temperature, and plot accordingly, and then find the average height of the graph [which is of course the area under the graph, divided by its width along the time axis…], what I mean is the quantity you would get by so doing”? These are mathematically equivalent quantities, sure, but there’s usually no need to use the latter perspective to explain the former; the former makes sense on its own.