Improper integrals debate

Here’s a simple Calc 1 problem:

Evaluate  \int_{-1}^1 \frac{1}{x}dx

Before you read any of my own commentary, what do you think? Does this integral converge or diverge?

image from illuminations.nctm.org

Many textbooks would say that it diverges, and I claim this is true as well. But where’s the error in this work?

\int_{-1}^1 \frac{1}{x}dx = \lim_{a\to 0^+}\left[\int_{-1}^{-a}\frac{1}{x}dx+\int_a^{1}\frac{1}{x}dx\right]

= \lim_{a\to 0^+}\left[\ln(a)-\ln(a)\right]=\boxed{0}

Did you catch any shady math? Here’s another equally wrong way of doing it:

\int_{-1}^1 \frac{1}{x}dx = \lim_{a\to 0^+}\left[\int_{-1}^{-a}\frac{1}{x}dx+\int_{2a}^{1}\frac{1}{x}dx\right]

= \lim_{a\to 0^+}\left[\ln(a)-\ln(2a)\right]=\boxed{\ln{\frac{1}{2}}}

This isn’t any more shady than the last example. The change in the bottom limit of integration in the second piece of the integral from a to 2a is not a problem, since 2a approaches zero if does. So why do we get two values that disagree? (In fact, we could concoct an example that evaluates to ANY number you like.)

Okay, finally, here’s the “correct” work:

\int_{-1}^1 \frac{1}{x}dx = \lim_{a\to 0^-}\left[\int_{-1}^{a}\frac{1}{x}dx\right]+\lim_{b\to 0^+}\left[\int_b^{1}\frac{1}{x}dx\right]

= \lim_{a\to 0^-}\left[\ln|a|\right]+\lim_{b\to 0^+}\left[-\ln|b|\right]

But notice that we can’t actually resolve this last expression, since the first limit is \infty and the second is -\infty and the overall expression has the indeterminate form \infty - \infty. In our very first approach, we assumed the limit variables a and b were the same. In the second approach, we let b=2a. But one assumption isn’t necessarily better than another. So we claim the integral diverges.

All that being said, we still intuitively feel like this integral should have the value 0 rather than something else like \ln\frac{1}{2}. For goodness sake, it’s symmetric about the origin!

In fact, that intuition is formalized by Cauchy in what is called the “Cauchy Principal Value,” which for this integral, is 0. [my above example is stolen from this wikipedia article as well]

I’ve been debating about this with my math teacher colleague, Matt Davis, and I’m not sure we’ve come to a satisfying conclusion. Here’s an example we were considering:

If you were to color in under the infinite graph of y=\frac{1}{x} between -1 and 1, and then throw darts at  the graph uniformly, wouldn’t you bet on there being an equal number of darts to the left and right of the y-axis?

Don’t you feel that way too?

(Now there might be another post entirely about measure-theoretic probability!)

What do you think? Anyone want to weigh in? And what should we tell high school students?

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**For a more in depth treatment of the problem, including a discussion of the construction of Reimann sums, visit this nice thread on physicsforums.com.

Cubic polynomials and tangent lines

Just read an article in the most recent NCTM Mathematics Teacher magazine called “Students’ Exploratory Thinking about a Nonroutine Calculus Task” by Keith Nabb. I really, really enjoyed this article. Maybe for some this isn’t new, but I didn’t know this fact:

Average two of the roots of a cubic polynomial. Draw a tangent line to the cubic at this point. Did you know it will always pass through the third zero?? Incredible!

Here’s a nice site that I just googled that goes through one proof. However, the charm of the article mentioned above is that there are many interesting proofs that students came up with, some of which are more or less elegant (brute force algebra with CAS, Newton’s Method, just to name two of the four strategies mentioned in the article).

I wish I could give you the whole article, but you have to have an NCTM membership to see it. Here’s the link, but you’ll have to log in to actually see it.

Integration by parts and infinite series

I was teaching tabular integration yesterday and as I was preparing, I was playing around with using it on integrands that don’t ‘disappear’ after repeated differentiation. In particular, the problem I was doing was this:

\int x^2\ln{x}dx

Now this is done pretty quickly with only one integration by parts:

Let u=\ln{x} and dv=x^2dx. Then du=\frac{1}{x}dx and v=\frac{x^3}{3}. Rewriting the integral and evaluating, we find

\int x^2\ln{x}dx = \frac{1}{3}x^3\ln{x}-\int \left(\frac{x^3}{3}\cdot\frac{1}{x}\right)dx

=\frac{1}{3}x^3\ln{x}-\int \frac{x^2}{3}dx

= \frac{1}{3} x^3 \ln{x} - \frac{1}{9} x^3 + c .

But I decided to try tabular integration on it anyway and see what happened. Tabular integration requires us to pick a function f(x) and compute all its derivatives and pick a function g(x) and compute all its antiderivatives. Multiply, then insert alternating signs and voila! In this case, we choose f(x)=\ln{x} and g(x)=x^2. The result is shown below.

\int x^2\ln{x}dx = \frac{1}{3}x^3\ln{x}-\frac{1}{12}x^3-\frac{1}{60}x^3-\frac{1}{180}x^3-\cdots

= \frac{1}{3}x^3\ln{x} - x^3 \sum_{n=0}^\infty \frac{2}{(n+4)(n+3)(n+2)(n+1)} +c

If I did everything right, then the infinite series that appears in the formula must be equal to \frac{1}{9}. Checking with wolframalpha, we see that indeed,

\sum_{n=0}^\infty \frac{2}{(n+4)(n+3)(n+2)(n+1)} = \frac{1}{9} .

Wow!! That’s pretty wild. It seemed like any number of infinite series could pop up from this kind of approach (Taylor series, Fourier series even). In fact, they do. Here are just three nice resources I came across which highlight this very point. I guess my discovery is not so new.

Why Calculus still belongs at the top

AP Calculus is often seen as the pinnacle of the high school mathematics curriculum*–or the “summit” of the mountain as Professor Arthur Benjamin calls it. Benjamin gave a compelling TED talk in 2009 making the case that this is the wrong summit and the correct summit should be AP Statistics. The talk is less than 3 minutes, so if you haven’t yet seen it, I encourage you to check it out here and my first blog post about it here.

I love Arthur Benjamin and he makes a lot of good points, but I’d like to supply some counter-points in this post, which I’ve titled “Why Calculus still belongs at the top.”

Full disclosure: I teach AP Calculus and I’ve never taught AP Statistics. However I DO know and love statistics–I just took a grad class in Stat and thoroughly enjoyed it. But I wouldn’t want to teach it to high school students. Here’s why: For high school students, non-Calculus based Statistics seems more like magic than mathematics.

When I teach math I try, to the extent that it’s possible, to never provide unjustified statements or unproven claims. (Of course this is not always possible, but I try.) For example, in my Algebra 2 class I derive the quadratic formula. In my Precalculus class, I derive all the trig identities we ask the students to know. And in my Calculus class, I “derive” the various rules for differentiation or integration. I often tell the students that copying down the proof is completely optional and the proof will not be tested–“just sit back and relax and enjoy the show!”

But such an approach to mathematical thinking can rarely be applied in a high school Statistics course because statistics rests SO heavily on calculus and so the ‘proofs’ are inaccessible. I’d like to make a startling claim: I claim that 99.99% of AP Statistics students and 99% of AP Statistics teachers cannot even give the function-rule for the normal distribution.

Image used by permission from Interactive Mathematics. Click the image to go there and learn all about the normal distribution!

Image used by permission from Interactive Mathematics. Click the image to go there and learn all about the normal distribution!

In what other math class would you talk about a function ALL YEAR and never give its rule? The normal distribution is the centerpiece (literally!) of the Statistics curriculum. And yet we never even tell them its equation nor where it comes from. That should be some kind of mathematical crime. We might as well call the normal distribution the “magic curve.”

Furthermore, a kid can go through all of AP Statistics and never think about integration, even though that’s what their doing every single time they look up values in those stat tables in the back of the book.

I agree that statistics is more applicable to the ‘real world’ of most of these kids’ lives, and on that point, I agree with Arthur Benjamin. But I would argue that application is not the most important reason we teach mathematics. The most important thing we teach kids is mathematical thinking.

The same thing is true of every other high school subject area. Will most students ever need to know particular historical facts? No. We aim to train them in historical thinking. What about balancing an equation in Chemistry? Or dissecting a frog? They’ll likely never do that again, but they’re getting a taste of what scientists do and how they think. In general, two of our aims as secondary educators are to (1) provide a liberal education for students so they can engage in intelligent conversations with all people in all subject areas in the adult world and (2) to open doors for a future career in a more narrow field of study.

So where does statistics fit into all of this? I think it’s still worth teaching, of course. It’s very important and has real world meaning. But the value I find in teaching statistics feels VERY different than the value I find in teaching every other math class. Like I said before, it feels a bit more like magic than mathematics.**

I argue that Calculus does a better job of training students to think mathematically.

But maybe that’s just how I feel. Maybe we can get Art Benjamin to stop by and weigh in!

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*In our school, and in many other schools, we actually have many more class options beyond Calculus for those students who take Calculus in their Sophomore or Junior year and want to be exposed to even more math.

** Many parts of basic Probability and Statistics can be taught with explanations and proof, namely the discrete portions–and this should be done. But working with continuous distributions can only be justified using Calculus.

Happy Mean Girls day

In my Calculus class today I showed just a short clip from this video (“the limit does not exist!”).

Apparently, completely and totally without my knowledge, I showed this video today, on October 3rd, which just happens to be Mean Girls Day. So..happy Mean Girls Day! (happy/mean…that sounds a little funny)

Kudos to the folks who put these videos together!