Why do we call integration "accumulation of change"?

Question

So in virtually all English-language calculus classes I have seen, we define integration as the "accumulation of change". And that makes sense to me intuitively, but when I think about it, I feel like accumulation of value makes more sense. Because if we take the change in a function $f:\mathbb{R}\mapsto\mathbb{R}\text{ s.t. } f(x)=k$ for $k\in\mathbb{R}$, then the "change" in $f$ at any $x$ is $0$. So accumulating it, we add $0$ with itself some number of times, right? Which will always be $0$. Yet, $$\int_a^b k\, dx$$ isn't always equal to zero...

I feel like accumulation of change makes sense to me but I can't put my finger on why it does. Can anyone try to explain?

Answer 1

Think of a car driving from one location to another. It can speed up or slow down, so its velocity can increase or decrease, but it is always changing position (except for when it comes to a stop). The total distance the car has traveled is the accumulation of all its changes in position, i.e. the integration of its velocity function.

Answer 2

The thing that you're integrating is the change that's being accumulated. When you want to calculate the cumulative change of $f(x)=k$, you integrate its rate of change, which you correctly identified as 0. So you should expect $f(b)-f(a)=\int_a^b 0\mathrm dx$, which is true.

When you integrate $f$ itself, you view $f$ as the rate of change of a different function $F$, and you expect $F(b)-F(a)=\int_a^b f(x)\mathrm dx$.

Answer 3

Viewing integration as the accumulation of something can be a reasonable way to think about it sometimes. But it's not possible actually to define integration just using a few words like that. All you can do with a description like "accumulation of change" is to hint at an idea that motivates the actual definition of an integral.

Here's an example of how an English-language calculus course introduces integration (from MIT Open Courseware Single Variable Calculus):

The definite integral tells us the value of a function whose rate of change and initial conditions are known.

The word "accumulation" does not occur in any form here.

Here's how integration is introduced in Paul's Online Notes:

In the past two chapters we’ve been given a function, $f(x)$, and asking what the derivative of this function was. Starting with this section we are now going to turn things around. We now want to ask what function we differentiated to get the function $f(x)$.

I didn't have to look very hard to find these examples. I simply knew that MIT has online versions of several of their undergraduate math classes called "open courseware", and that Paul Dawkins, who teaches calculus at Lamar University, has put a very useful set of course notes on the Web called Paul's Online Notes. These were literally the first two places I looked to see how "English-language calculus classes" present integration.

Next, I tried looking for "accumulation of change" explicitly. The first search hit was Khan Academy:

The definite integral is an important tool in calculus. It calculates the area under a curve, or the accumulation of a quantity over time.

That's closer to your "accumulation of value" than "accumulation of change".

You can certainly find the phrase "accumulation of change" in many places online that explain integration, but I think it is simply not true that "virtually all English-language calculus classes" define (or even initially describe) integration in exactly those terms.

The problem with attempting to describe integration using English words such as "accumulation" is that words in English tend to have multiple meanings. "Accumulation", in particular, is a noun that can describe both the process of accumulating something and the result of accumulating something. The word "change" is also ambiguous. Sometimes we use it to mean the difference between what something was before and what it is after a process. Sometimes we use it to refer to the process itself.

To address your specific example of a constant function, defined by $f(x) = k$, what is the "change"? If the "change" is the derivative $f'(x)$, then an "accumulation of change" might be represented by

$$ \int_a^b f'(x)\,\mathrm dx, $$

and indeed this is always zero since $f'(x) = 0$. But when we write a generic definite integral,

$$ \int_a^b f(x)\,\mathrm dx, $$

we're only interested in integrating $f(x)$ itself, and the derivative of $f$ doesn't really come into the picture.

In short, don't treat phrases such as "accumulation of change" as mathematical definitions that you can reason about as formally and precisely as you attempted to do in the question. Such phrases are not meant to be used that way.

Also, if you really are interested in how integrals are defined, and all you're finding is "accumulation of change", broaden your search so you find some of the many texts that explain integration in other ways.