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.”

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