Background: I taught BC Calc for the first time last year, as an independent study – just one student. I leaned pretty heavily on the book’s presentation of the material. We use Larson and Edwards, 9th edition – pretty much your standard calculus book.
That particular student had, in a previous class, asked for more proofs. That’s a request I will leap at with both hands and latch on to with a death grip; we proved almost everything. The book has plenty of proofs, and I was scrambling to get the material re-learned and presentable, so I was quite willing to take the book’s proofs when they made sense.
The problem with that, of course, is that textbooks aren’t written for high school students. Have you ever looked at the proof of the Shell Method in Larson? Probably not, so I’ll tell you: it’s a nice idea, and they do a decent job explaining it. But it’s heavily algebraic. And I don’t know about your kids, but my kids don’t find algebra intuitive. In fact, I don’t find algebra intuitive. An algebraic proof will convince me that something is true, but it doesn’t help me understand why it’s true. I’d rather my kids understand something than just believe it as a revelation from on high.
This year, I’m teaching BC again. We start the year where we left off with AB – area and volume, which means we got to the Shell Method on day 2 of class. As I was looking over my notes, I realized just how rough that algebraic proof was going to be.
Quick recap: the shell method of volume is basically a Russian stacking doll, but with cylinders instead of dolls. You take infinitely many, infinitely thin outer slices of cylinders and add up their volume to get the volume of your solid of revolution. It turns out the that formula for the volume of a cylindrical shell is the circumference times the height times the width of the shell (which turns into dx in your integral, because it’s infinitely thin.)
The algebraic proof finds the volume of the entire cylinder, the volume of the hole you cut away to get the shell, and then subtracts to get the volume of the shell. Simple, straightforward, and totally obfuscated by the algebra.
Ten minutes before class started, it hit me that length*height*width is the volume of a rectangular prism. And if you take a cylindrical shell and slice it open on one side, it unrolls to a rectangular prism, where the circumference is the length. So instead of going through a bunch of messy algebra, I took a piece of paper and rolled it into a cylindrical shell. Then I unrolled it into a rectangular prism. Look, guys, same volume! Circumference becomes length! Volume formula that you learned in sixth grade? Great, let’s move on.
Why did I not see this before?
Is this the standard proof? Does everyone teach it this way and I’m just slow to the party?
If yes, then for goodness’s sake, why was that not in a textbook? That’s so much more intuitive then a half-page of algebraic manipulations! If no, then why not? Is there something I’m missing here?