Simple But Evil #2 — Improper Integrals

In this edition of Simple But Evil, I’ll talk improper integrals, limits [naturally!], and some silly things that happen. By and large, most Calculus texts that I’ve seen do a good job of giving a mix of improper integrals that challenge students’ notions of basic arithmetic with infinities.

As almost any math professor will tell you, a lot of students default to saying that \(\frac{\infty}{\infty} = 1\) or \(\infty – \infty = 0\) or \(0 \cdot \infty = 0\) or \(\infty^{0} = 1\) or \(1^{\infty}\) = 1. And why not? Certainly we have that \(\infty + \infty = \infty\) and that \(\infty \cdot \infty = \infty\). Since some finite arithmetic rules match the rules for arithmetic with infinities, then by an abuse of induction, should all rules match? Of course, this isn’t so.

Part of the problem is that the notion of finite arithmetic hasn’t been separated from arithmetic with infinities. We’ve kind of left the infinities alone until we hit pre-Calc or depending on the course, sometimes we discuss this in Algebra when talking about “end behavior” of polynomials. In either case, by the time they’ve reached improper integrals, students have had exposure to working with limits and to working with the arithmetic of infinities. But alas, things will remain mixed up.

Without further ado …

Symmetry

I’ve found that using a sequence of symmetrical integrals is a good way to get the discussion going about \(\infty – \infty\). There are other ways to have the conversation [appeal to mathematical theorems, eg], but when I teach I tend to take things a little slow for the stuff that is counterintuitive. I want to first allow for the counterintuitive thoughts to get out in the open before I address them. Handing out the requirement that in order for \(\int_{-\infty}^{\infty}f(x) dx\) to converge we must have that \(\int_{c}^{\infty}f(x) dx\) and \(\int_{-\infty}^{c}f(x) dx\) both converge, I think robs them of the thought development process. The counterintuitive notions will exist even if we lay down the theorem. Thus, I think it’s best to deal with their notions before putting forth the theorem.

Here’s how I start — simple, but mildly evil.

  1. \(\displaystyle \int_{-1}^{1}x dx\)
  2. \(\displaystyle \int_{-10}^{10}x dx\)
  3. \(\displaystyle \int_{-100}^{100}x dx\)
  4. \(\displaystyle \int_{-\infty}^{\infty}x dx\)

Naturally, the first three integrals evaluate to zero, but the fourth? Part of the problem is that students want to do this
\(\displaystyle \int_{-\infty}^{\infty}x dx = \lim_{b\rightarrow\infty}\int_{-b}^{b}x dx\) resulting in \(b^{2}-b^{2}\). While this is understandable, it isn’t correct. There are some hand-wavy ways to counter this set up, one of which is to question why we must approach \(+\infty\) and \(-\infty\) at the same “rate” (a la \(b\)).

What happens in this conversation is that students will appeal to the symmetry and say “we have the same area on the negative side as on the positive side”. Now, I bring in a counterexample using logic similar to theirs.

  1. \(\displaystyle \int_{0}^{1}3x^{2} dx + \int_{-1}^{0} 2x dx\)
  2. \(\displaystyle \int_{0}^{\infty}3x^{2} dx + \int_{-\infty}^{0} 2x dx\)

The definite integrals above have the same area with opposite signs. But the improper integrals?

I argue that the subtraction of the improper integrals should also be zero since we have the same area (but opposite signs) on the negative side as on the positive side. I have an area of infinity from \([0, \infty)\) and an area of \(-\infty\) from \((-\infty,0]\). For most students this is not believable since clearly “the area under \(3x^{2}\) is larger than the area under \(2x\)”.

The kicker here is that since I have an infinite area to play with, I can always cancel out the area under \(3x^{2}\) from \([0,a]\) by choosing \(b\) so that the area under \(2x\) matches. In other words, $$\int_{0}^{a}3x^{2} dx + \int_{-b}^{0}2x dx = 0$$ for \(b = a^{\frac{3}{2}}\). From here we can have a deeper conversation about where we have freedom of choice with our bounds of integration. The upper and lower bounds of \(+\infty\) and \(-\infty\) are not tied to together by a single variable that we let go to infinity.

Another side conversation is about which symmetry the students are focused on. They are focused on symmetry about \(y\) and a particular choice for how we take the limit. But what of this?
$$\displaystyle \lim_{a\rightarrow\infty}\Bigg(\int_{0}^{a}3x^{2} dx + \int_{-a^{\frac{3}{2}}}^{0} 2x dx\Bigg)$$

Convergent double-sided improper integrals shouldn’t depend on how I tend towards infinity and hence, loosely, why we require that both \(\int_{c}^{\infty}f(x) dx\) and \(\int_{-\infty}^{c}f(x) dx\) must converge independently in order that \(\int_{-\infty}^{\infty}f(x) dx\) converge.

There is, of course, a little hand-waving, but I’m more inclined to get them to first accept that they cannot default to \(\infty – \infty = 0\) and then we can revisit the formalities. For what it’s worth, proofs of some theorems tend to be omitted in Calculus texts and so students are left to accept the theorem as fact without letting go of their intuition. We have to fix the intuition first.

Go The Extra Mile

A standard problem is to show that \(\displaystyle \int_{0}^{\infty}e^{-x} dx\) converges. I like to then ask, find all values \(p\) so that \(\displaystyle \int_{0}^{\infty}e^{px} dx\) converges. The lesson about solving inequalities effectively stops at pre-Calculus. There’s some stuff with the Squeeze theorem and then a revisit with tests of convergence for series, but by and large, students end up forgetting how to work with inequalities. This is a good place to work off some of that rust.

Continuing with the exponential, this problem is also worth giving:

Find a value of \(p\) so that \(\displaystyle \int_{0}^{\infty}e^{px} = 3\).

A Variation On A Theme

Related to “Go The Extra Mile”, I’ve occasionally tried this. We are given that \(\displaystyle \int_{0}^{\infty}e^{-x^{2}} = \frac{\sqrt{\pi}}{2}\). What then is \(\displaystyle \int_{0}^{\infty}e^{-a^{2}x^{2}}\). Simple but evil — students had better remember how \(u\)-substitution works.

Central Discontinuity

Most Calculus texts that I’ve worked with do a good job of giving problems where the integrand has a discontinuity between the bounds of integration. So, this is more annoying than evil.

$$\int_{-1}^{1}\frac{dx}{x}$$

Setup For Series

Students will get their taste of alternating series a little after the business of improper integrals is done. I like to do a lot of foreshadowing when I teach a course. These three problems are worth discussing. They don’t show up enough in the improper integrals part of Calculus II.

  1. \(\int_{0}^{\infty}\cos(x) dx\)
  2. \(\int_{0}^{\infty}e^{x}\cos(x) dx\)
  3. \(\int_{0}^{\infty}e^{-x}\cos(x) dx\)

Some helpful supplementals are graphs of the integrands and / or the results of the integrals when the bounds of integration are finite. Varying the bounds helps to convince students that one of the three integrals is converging and the other two are diverging. These also serve as a reminder that divergence doesn’t mean “tending to infinity”. That’s another conflation that rears its ugly head in the discussion of sequences and series. It’s good to get that started here as well.

Other Thoughts

There are a lot more things to discuss. Students want to “plug in \(\infty\)” when working with limits — a bad habit that probably got started in Calculus I and the introduction of limits. Students also want to “start at negative infinity and to go positive infinity” when understanding the notation of \(\int_{-\infty}^{\infty}\) since this is how we explain Riemann integration and working the bounds of integration. It’s an understandable extension. But lost at the heart of this is that infinity is not a number. It’s worth saying this repeatedly. Of course, some of you may be quick to point out stereographic projection where we could think of infinity as a point that’s reachable, but that instance is so far removed that sticking with “infinity is not a number” and “you can’t get to infinity” is a good mantra. Appending with “in this class” is also a reasonable way to keep things long-term consistent.

On a personal note and it’s something that I never bring up in class, but how I first understood arithmetic with infinities was at a theological level. It’s cheesy, but as a kid, I remember it made sense. It’s easy to point out contradictory behavior in the existence of an absolutely omnipotent god. Can God create an object He can’t move? If possible, then we don’t have omnipotence, since the object can’t be moved by a supposedly omnipotent being. If not possible, well, clearly, there’s something that the omnipotent being can’t do. Of course, a lot of this at a theological level, comes down to semantics of what type of omnipotence we’re talking about. But as a kid, when I was thinking about infinities, I was often drawn to the idea of God and omnipotence. I had come to accept that the contradictions are only contradictions because we’re bound by our finite logic. In fact, in my mind, God can indeed create an object He can’t lift. We’re in the world of infinite power, let us not constrain it by our finite logic. A contradiction in the world of mortals means nothing in His world. Hence, \(\infty – \infty\) doesn’t have to be zero, because why should the infinities obey the laws of finite arithmetic? They have their own laws. As such, I was ok with consistent behavior of \(\infty + \infty\), \(\infty\cdot\infty\), and \(\infty^{\infty}\) and that the other interactions would not give consistent results since it would be equivalent to solving “if the immovable object meets the unstoppable force, who wins?”.

Next week will be Simple But Evil for L’Hospital’s rule and I’ll look to make this a weekly series. In the meanwhile, do let me know your thoughts! If you have a topic you want me to discuss, just ask in the comments or shoot me a tweet or email!

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2 thoughts on “Simple But Evil #2 — Improper Integrals

  1. Jim Doherty

    You got me at Go the Extra Mile. I also LOVE the discussion comparing 3x^2 and 2x. My BC gang had a similar symmetry conversation regarding integrals that span over vertical asymptotes. The appeal of comparing negative and positive infinite areas was pretty powerful. Much more so than in the days before we could fire up GeoGebra or Desmos to look at lovely graphs, that’s for sure. Thanks for starting this series, I shared it with both of my Calculus colleagues here.

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