Does this explanation of integration and the Fundamental Theorems of Calculus make any sense?

First, Sue Pemberton (Pure Mathematics 1 Coursebook, 2018, Cambridge University Press) introduces

integration as the reverse process of differentiation ...

$$\int x^3 \mathrm d x$$ is called the indefinite integral of $$x^3$$ with respect to $$x$$.

Next, she introduces definite integrals as simply mechanically taking an indefinite integral and evaluating it by plugging in the two limits.

Next, she gives this explanation:

The above explanation strikes me as being complete nonsense. In particular, it is a completely nonsensical explanation of the Fundamental Theorems of Calculus. Why is that

As $$\delta x \to 0$$, then $$A\to \int^5_2 y \mathrm d x?$$

But since this book was published by Cambridge University Press, it's quite possible that I'm mistaken.

How might the above explanation make any sense?

• I wonder if there is more in the book, that isn't shown here? Without somehow using the fact that derivative gives rate of change, this is missing the heart of the problem. Commented Jul 17 at 2:58
• There is a tradition of certain "Calculus Lite" books that give the facts of calculus without giving any explanations. This seems to be one of them. Commented Jul 17 at 8:57
• Interesting to note that the "rectangular strips" in that figure aren't rectangles.
Commented Jul 17 at 12:22
• Yeah, this is at best a poor explanation. As a fledgling math major it would have been exceedingly unhelpful to me. Moreover, if you want to cheat, there are easier arguments. Draw a picture of $A(x)$ draw a picture of $A(x+\Delta x)$. Argue $\Delta A = f(x) \Delta x$ thus $\Delta A/ \Delta x = f(x)$. That gives you FTC I. Then FTC II is easy to derive from there. The tricky part is the existence of an antiderivative, but the existence of an area function almost immediately suggests and antiderivative once you see it. Commented Jul 17 at 12:33
• I don't see a claim in the quoted text that this justifies either of the fundamental theorems. It seems like just an approximate definition of the definite integral, skipping over the formalities involved with the Riemann sum definition. Commented Jul 17 at 14:30

I agree that the explanation is incomplete and it seems like a major conceptual leap is left to the reader.

I've written about this before and this is how I made the jump. (Note that in the explanation below, $$F$$ is an antiderivative of $$f.$$)

• This is a nice derivation of FTC II ;$\int_a^b f(x)dx = F(b)-F(a)$. The subtle thing about FTC I ($\frac{d}{dx} \int_a^x f(t)dt = f(x)$) is that it gives the existence of the antiderivative in the form of the area function $\int_a^x f(t)dt$. That is the sneaky thing assumed here, that given $f$ there exists $F$. Otherwise, this is far more direct than the typical argument for $\int_a^b f(x)dx = F(b)-F(a)$. Commented Jul 17 at 12:29

In this explanation, context is everything. In a comment below the question (which I am reproducing here in case it is ever deleted), David Roberts notes

the book [1] is designed to go along with this specific curriculum: [2] which as you guessed is high school level.

The intended audience of this text is high school students (albeit ones who, if my understanding of the British education system is correct, are hoping to earn college credit for their work). To put it in a context with which I am more familiar, this looks like an AP calculus text. Such texts are generally written to prepare students for standardized exams which, if they are completed up to a certain level, can be used to waive college classes (and sometimes to earn college course credit). In my experience, these exams focus more on computation than on theory—the emphasis is on building computational skills which will be useful to physics and engineering students.

In this context, the explanation is not unreasonable, and is perfectly adequate. It is entirely correct to state that (1) the notation

$$\int f(x) \,\mathrm{d}x$$

represents an antiderivative of $$f$$ (i.e. a function $$F$$ such that $$F' = f$$), that (2) the area under $$f$$ can be approximated by the Riemann sum

$$\sum y \,\delta x$$

(where $$y$$ corresponds to the height of the curve in some subdivision of the interval $$[a,b]$$, and $$\delta x$$ is the uniform width of the subdivision), and that (3) as $$\delta x \to 0$$,

$$\sum y\, \delta x \to \int_{a}^{b} f(x) \,\mathrm{d}x = F(b) - F(a).$$

The first statement is the definition of some notation, the second is a somewhat vague explanation of how the definite integral is defined, and the third is a statement of the Fundamental Theorem of Calculus (part 2, in most of the texts I have taught out of, where part 1 asserts that $$\frac{\mathrm{d}}{\mathrm{d}x} \int_{0}^{x} f(t)\,\mathrm{d}t=f(x)$$.) This is sufficient information for students to compute definite integrals, while giving only the barest thumbnail sketch of the theory.

Each of these three statements can be made more rigorous. In a more advanced class (e.g. a stronger college calculus class, an "advanced calculus" class, or an elementary class on real analysis), the definitions would likely be presented a bit more precisely, and there would be at least some motion towards a proof of the Fundamental Theorem of Calculus. But the fact that these statements can be made more rigorous does not imply that every student needs them to presented in that more rigorous form.

Again, note that the goal of these notes is to give high school students a rough sketch of the theory so that they can compute in the context of an exam which will give them college placement and/or credit. It is entirely reasonable to ask whether or not this is an appropriate goal, but in that context, the approach is... fine.

That being said, I really dislike the second figure:

The text explains that the area under the curve can be approximated by rectangles, but the figure does not show this. The figure shows the area being decomposed into vertical strips which are most definitely not depicted as rectangles. This, in my opinion, kind of sloppy.

• The A and AS levels would be taken in England and Wales by pupils leaving secondary school, so about 18 years old. Entry to university depends on those grades (and other things) but when you get there the slate is wiped clean. Commented 17 hours ago
• I agree that you have to adjust to the level of audience when explaining things, but, IMHO, the less the audience knows, the more careful you need to be with your words not to say some nonsense and the more time you need to think about how to accurately convey the meaning without going into technical details. From this perspective, this explanation is totally inadequate. Downvoted. :-) Commented 16 hours ago
• I'm surprised that the (non)rectangles bother you. If the text is for engineers and physicists, don't you think the error is small enough that they're basically rectangles? They'll see actual rectangles in a better calculus course :) Commented 14 hours ago
• @Thierry If you say "rectangles", you should draw rectangles. The Big Idea™ of calculus is that the errors are small, and disappear in the limit. But there are still errors there, which should be conveyed in the picture. Commented 14 hours ago
• My big issue is this: Pemberton simply and very mysteriously claims, As $\delta x \to 0$, then $A\to \int^5_2 y \mathrm d x$. (Are students somehow supposed to know where this comes from?) Nowhere does she state anything like the third is a statement of the Fundamental Theorem of Calculus. (And in addition, what is this "Fundamental Theorem of Calculus"? Are students supposed to already be familiar with it?) You are using what you already know to justify what she's written. I'm trying to come from the perspective of a student reading all this for the first time. Commented 6 hours ago