How I Teach Calculus: A Comedy (Trigonometric Substitution)

They said it couldn’t be done! They said I was crazy, that I had finally bit off more than I could chew. (I have an exceptionally large mouth, within which can fit things like, say, an entire apple, whole whoopie pies, most things on White Castle’s menu)

The seniors from last year have come back, and they have suggested I teach trigonometric substitution during my tour of integration techniques. I floated the idea around and received death threats, brimstone (disguised as fruit cake!), and an inexplicable stuffed rabbit covered in blood on my pillow. In education, these are generally signs to plow onward.

Why the guff? Well, trig sub, as it’s called in the biz, is notoriously confusing. It usually involves a thousand different steps each of which requires the most careful attention by even the most metered trigonometric minds. Triangles with, like, more than one variable, and stuff.

Proceduralism Makes Me Want To Die

I make up words. Proceduralism is the soon-to-be defunct addiction that most math teachers have, especially those who like to use words like “accelerated” and “algebra” in the same sentence. Proceduralism is the practice of teaching step-by-step rote processes that result in nothing but correct answers. This can be applied to many beautiful things with Dorian-Gray-like results: I’m looking at you Geometry (Proofs), Pre-Calculus (Induction), and you, you, Calculus, how dare you water down Optimization.

I imagine myself as sort of Beowulf-Gandalf-Odysseus-John Goodman hybrid. I will not stand for proceduralism in my classroom, and Trig Sub represents my Grendel-Sauron-Siren-Rosanne arch-nemesis. So easy to teach as a bunch of steps, so meaningless when done that way. Extensibility, thy name shall be math education. If they can’t recognize it outside of the context of your textbook, you have taught nothing.

Here’s what makes it even worse: some students have been passed up to me that can only handle proceduralism. I don’t blame anyone, but crikey, it sure does make things difficult. These students require careful attention. Their brains have not yet developed to the point of handling the truth and beauty (not even strangeness or charm yet!) that is higher mathematics, but yet there they sit in my calculus class. They must be taught, and it must be at a procedural level. I trust that they will review and make the connections later in life, but how do I differentiate so as to serve those students that are both willing and able to take the heat?


I love ellipses. Let’s put it this way: I’m a junior constable in Kepler’s Law enforcement. Personally, I think the circle gets too much credit. It’s boring. Ellipses, so ovoid, so, so . . . squishy. They need a little love. So we begin here:

What’s the area of an ellipse?

Circle’s are easy. They have that down pat. An ellipse? Now that’s crazy talk. Some venture that it’s:

[latex size=4] A = \pi a^2 – \pi b^2[/latex]

I like their thinking, but what’s the motivation? a represents the major axis length, and b the semi-major axis length. By how can you be sure? Graph it? Ok:

CBS? Blue Circle Area - Red Circle Area = Green Ellipse Area?

This is exactly the kind of thing I like to have bubble up from the class. It’s probably not correct, but who cares; now we have a mission. What is the area of the ellipse, and how can we find it reliably so that we can test our intuition?

I won’t reproduce the derivation of the area of an ellipse here. Suffice it to say that the math is fun and genuinely requires a new technique (the aforementioned trigonometric substitution; all good Scrabble words!)

The key events are:

1. Simplify the integral so that you’re only calculating a quarter of the area, which has the nice side effect of ignoring any implicit behavior. As:

[latex size=4]A = \frac{4b}{a} \int^a_0\sqrt{a^2-x^2}dx[/latex]

2. Notice that this integral is quite obstinate, and refuses even the most stalwart algebraic substitutions and attempts at integration by parts. It requires us to know and love Pythagoras, which is the easiest way to produce/excise such unruly roots.

3. Much cajoling later, we end up with the following integral (don’t believe me, here’s the work):

[latex size=4] A=4ab\int^{\pi/2}_0 [\cos{\theta}]^2d\theta[/latex]

4. The integral of cosine squared requires some additional thought, and usually another day in class, but can be easily worked using an identity. Leaving you, finally, with:

[latex size=4] A = \pi a b [/latex]

Anticlimactic? Beautiful? What’s even crazier is that when a=b (i.e. a circle) you get pie arr squared. Yay!

Let’s check against our initial guess. The area of the following ellipse:

[latex size=4] \frac{x^2}{9}+\frac{y^2}{4} = 1[/latex]

[latex size=4] A = 6\pi[/latex]

And, from our initial conjecture:

[latex size=4] A = \pi a^2 – \pi b^2[/latex]

[latex size=4] A = 5\pi[/latex]

Doesn’t match. Hmm. Would it ever get close? Is there any logic to our guess? Who knows, but the creativity here is what I’m after.

The kids dig into this. There’s something about developing the formulas that really makes them feel like they finally own a part of their muddled history of math education.

Oh, and I recorded a bunch of Christmas music.