My Favorite One-Liners: Part 101

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

I’ll use today’s one-liner when a choice has to be made between two different techniques of approximately equal difficulty. For example:

Calculate $\displaystyle \iint_R e^{-x-2y}$, where $R$ is the region $\{(x,y): 0 \le x \le y < \infty \}$

There are two reasonable options for calculating this double integral.

• Option #1: Integrate with respect to $x$ first:

$\int_0^\infty \int_0^y e^{-x-2y} dx dy$

• Option #2: Integrate with respect to $y$ first:

$\int_0^\infty \int_x^\infty e^{-x-2y} dy dx$

Both techniques require about the same amount of effort before getting the final answer. So which technique should we choose? Well, as the instructor, I realize that it really doesn’t matter, so I’ll throw it open for a student vote by asking my class:

Anyone ever read the Choose Your Own Adventure books when you were kids?

After the class decides which technique to use, then we’ll set off on the adventure of computing the double integral.

This quip also works well when finding the volume of a solid of revolution. We teach our students two different techniques for finding such volumes: disks/washers and cylindrical shells. If it’s a toss-up as to which technique is best, I’ll let the class vote as to which technique to use before computing the volume.

My Favorite One-Liners: Part 60

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

I’m a big believer using scaffolded lesson plans, starting from elementary ideas and gradually building up to complicated ideas. For example, when teaching calculus, I’ll use the following sequence of problems to introduce students to finding the volume of a solid of revolution using disks, washers, and shells:

• Find the volume of a cone with height $h$ and base radius $r$.
• Find the volume of the solid generated by revolving  the region bounded by $y=2$, $y=2\sin x$ for $0 \le x \le \pi/2$, and the $y-$axis about the line $y=2$.
• Find the volume of the solid generated by revolving the region bounded by $y=2$, $y=\sqrt{x}$, and the $y-$axis about the line $y=2$ .
• Find the volume of the solid generated by revolving the region bounded by $y=2$, $y=\sqrt{x}$, and the $y-$axis about the $y-$axis .
• Find the volume of the solid generated by revolving the region bounded by $x=\sqrt{2y}/(y+1)$, $y=1$, and the $y-$axis about the $y-$axis .
• Find the volume of the solid generated by revolving  the region bounded by the parabola $x=y^2+1$ and the line $x=3$ about the line $x=3$.
• Water is poured into a spherical tank of radius $R$ to a depth $h$. How much water is in the tank?
• Find the volume of the solid generated by revolving the region bounded by $y= x^2$, the $x-$axis, and $x=4$ about the $x-$axis.
• Find the volume of the solid generated by revolving the region bounded by $y= x^2$, the $x-$axis, and $x=4$ about the $y-$axis.
• Repeat the previous problem using cylindrical shells.

In this sequence of problems, I slowly get my students accustomed to the ideas of horizontal and vertical slices, integrating with respect to either $x$ and $y$, the creation of disks and washers and (eventually) cylindrical shells.

As the problems increase in difficulty, I enjoy using the following punch line:

To quote the great philosopher Emeril Lagasse, “Let’s kick it up a notch.”

Volume of solid of revolution

In Calculus I, we teach two different techniques for finding the volume of a solid of revolution:

• Disks (or washers), in which the cross-section is perpendicular to the axis of revolution, and
• Cylindrical shells, in which the cross-section is parallel to the axis of revolution.

Both of these could be expressed as either an integral with respect to x or as an integral with respect to y, depending on the axis of revolution. I won’t go into a full treatment of the procedure here; this can be found in places like http://www.cliffsnotes.com/math/calculus/calculus/applications-of-the-definite-integral/volumes-of-solids-of-revolution or http://mathworld.wolfram.com/SolidofRevolution.html or http://en.wikipedia.org/wiki/Disk_integration or http://en.wikipedia.org/wiki/Shell_integration.

A natural question asked by students is, “If I have the choice, should I use disks or shells?” The correct answer, of course, is “Pick the method that gives you the easier integral to compute.” But that’s not a very satisfying answer for novice students who’ve just been exposed to integral calculus. So, over the years, I developed a standard reply to this query:

That’s an excellent question, and it’s one of the classic conundrums faced by mankind over the years.

Should I choose Coke… or Pepsi?

McDonald’s… or Burger King?

Ginger… or Mary Ann?

Disks… or shells?

The answer is, it just takes a little practice and experience to determine which technique gives you the easier integral.

If you don’t get the cultural reference, here’s a reminder. As of 10 years ago, I could still tell this joke to college students and still get smiles of acknowledgement. But, given the passage of time, I’m not sure if this same joke would fly college students now.