This WitCH is from Cambridge’s 2020 textbook, Mathematical Methods, Unit 1 & 2. It is the closing summary of Chapter 21A, Estimating the area under a graph. (It is followed by 21B, Finding the exact area: the definite integral.)
We’re somewhat reluctant about this one, since it’s not as bad as some other WitCHes. Indeed, it is a conscious attempt to do good; it just doesn’t succeed. It came up in a tutorial, and it was sufficiently irritating there that we felt we had no choice.
Drawing the rectangles would be a good start…
The formulas, as written are not going to be much use to 99% of Methods 1&2 students and probably ignored by 90% of Methods 1&2 teachers (possibly in favor of diagrams with the rectangles already drawn)
Other irritating parts: it says *determine* rather than *estimate* the area under the curve and there is no mention of how the estimate improves as n increases.
Thanks, RF. All good points. The one which *really* pissed me of, and which for me made it a WitCH, was the use of “determine” instead of “approximate”. That undermines the entire purpose of the section. (Which they don’t follow through on anyway …)
Look, it’s not all bad, but it could be a lot better. I don’t know how much explanation has been given previously in the text, given that this is supposed to a summary. Adding to Red Five’s comments:
* In the top line, x_0 is written as a, but x_n is not written as b.
* The diagram is sloppy and the positions of the outer ordinates could be far better placed.
* Left and Right rectangles are all very well, but it is a wasted opportunity not to point out that the Trapezoidal rule is an average of the two rectangle sums.
* It’s also a wasted opportunity not to mention the mid point rule: summing the rectangles of height f((x_k+x_{k+1})/2) .
* The rules can be used with functions that are not continuous: as long as they are bounded above and below, and defined everywhere in the interval. This is possibly the biggest mathematical error: made worse by the use of an Important Red Bold font.
* “Estimate” is surely the wrong word? The correct word should be “approximate”. Red Five has correctly pointed out that it should be mentioned that accuracy increases with n: for advanced students it may be worth pointing out also that the error can be exactly defined in terms of ((b-a)/n)^2.
If I can think of anything else, I’ll write again!
Alasdair the annoyed
A very thorough analysis, Alasdair! I also agree that the continuity bit is a bad error – it’s obvious that you can approximate the area under piece-wise DIScontinuous functions (bounded above and below, of course) and students would certainly have met examples of such functions.
I also agree that some comment on accuracy is appropriate, and it could have been mentioned that the average of the left- and right-rectangles approximations is exactly equal to the trapezoidal approximation. The proof is simple (but as you pointed out, it’s hard to know how much explanation was previously given in the text, given that this is supposed to a summary.
And if you’re going to talk about this stuff, why not mention Simpson’s Rule?
I also dislike the diagram – I think it’s lazy that the summary doesn’t include diagrams explicitly showing all three approximations and hence the ‘intuition’ behind the formulae.
Thanks, JF. There’s a real question of what the section is for. If it is just to motivate the following (so-so) section on area/integrals as Riemann sums, then the trapezoid rule has no business there. But if we’re really interested in those approximations, then the approximations should be discussed and related.
Thanks, Alasdair. A very good summary, and I agree that there’s *some* elements of the excerpt that are OK. But, mostly it is clumsy and pointless.
I agree, the “continuous function” in neon lights is particularly annoying. Given we’re just approximating here, the damn function can be absolutely anything. The emphasis on “continuous function”, together with the erroneous use of “determine”, is fundamentally confusing finite approximation with the equality (or definition) in the limit.
How can a function not be continuous? I’m joking of course, but do students in MM1&2 deal with discontinuous functions? (Maybe they do – I do not teach this subject.)
It is true that these methods can be used for “any continuous function” on a finite interval even though they can be used more widely.
I don’t see much difference between “estimate” or “approximate” – but “determine” is not the correct word here.
Quite often in text books, the introductory material in a chapter is loose and sloppy and quite different from the summary at the end of the chapter, especially with respect to definitions. This is not helpful to the careful reader, and we must remember that many teachers have little support beyond the text book.
The article below is enjoyable to read.
https://www.quantamagazine.org/james-maynard-solves-the-hardest-easy-math-problems-20200701/
Hi, Terry. In the next section the text gives the sense of a proof of the fundamental theorem of calculus; for that they need the intuition of continuity. But in this approximation section, they needn’t refer to the properties of the function whatsoever; the “approximations” can be applied to *any* function. How good those approximations are, or whether there is even something there to approximate, is still a question, but not a question they address or can be expected to address.
The estimation-approximation thing is not a major error, but they’ve chosen the wrong word. The use of “determination” is, as you suggest, a much greater issue.
The article was a very good read. Thanks.
Thanks for the article Terry. I had not heard of Maynard before.
A very magnanimous gesture from Tao to withhold publishing his work – of course, he had the luxury of doing that given his status and position. Does anyone have a bad word to say about him? Just seems like an all round good guy doing great things for the discipline and the academic community.
I have a bad word for everybody. But no, I can’t think of an iota of criticism to throw at him.
Hi,
The bounded function can be almost anything… but not everything for Riemann sums to work
Eg f(x) = 1 if x is rational and 0 otherwise has no unique Riemann Integral on [0,1] say
It would be nice if the summary mentioned the relative errors of the methods as n increases
And as a result why Simpson’s method is to be preferred over the methods described
Steve R
Hi, Steve. As I indicated in other replies, the function in this approximation section can be absolutely anything. Sure, the crazy function you suggest has no (Riemann) integral, but in this section that is neither here nor there, and of course an obscure point for a Year 11 treatment. In this section, and in general, the students don’t know that *any* function has a (Riemann) integral.
As for the error estimates, if the text were really interested in approximations then something about error size would of course be appropriate. But they’re not really interested in this: they’re just setting things up for the super-quick discussion of Riemann sums in the section that follows. That’s what makes including the trapezoid rule so ridiculous.
Marty ,
Ok for year 11’s then I suppose but I don’t like the continuous in bold and I haven’t seen the text ….
Perhaps it was just getting at taking the midpoint estimate only then as a third option for doing the numerical integration.
Steve R
The continuous not in bold is a needless distraction. The bold makes it utterly ridiculous.