Unit 10: Applications of the integral

I introduced this method with only one example in Video 10.1. This question simply asks students to write the general formula for an arbitrary function.

Students find this easy — assuming they have watched the video.

This method is normally called the “washer method” in textbooks when the region is bounded between two curves. I also refer to it as the “disc method” or as “the carrot method”, due to the prop I use in Video 10.1.

This is a standard example to practice computing the volume of a solid of revolution by slicing it into discs.

This example has various advantages over others:

• We are asking to compute the volume of a sphere, rather than an artificial example.

• We are not giving them the equation of a function to rotate around the $x$-axis. They have to figure out how to model the sphere as a solid of revolution and thus they cannot simply use a formula.

• It is satisfying to finally prove the equation for the volume of a sphere after years of using it without knowing where it came from.

This is one of the simplest examples of a solid whose volume can be computed by “slicing it like a carrot”, but that is not a solid of revolution. They have to start from scratch: they cannot use any of the previous formulas, and they have to fully understand the “slicing method” to set this up as integral. For these reasons, this activity is probably the best activity in the unit.

I am particular careful not to waste this activity. I give students enough time, and I make them do it, rather than wait for me to do it. I usually have students think for a while, then discuss together a bit of the set up, then they continue working, then we discuss a bit more, and so on.

In both Video 10.1 and Video 10.2 I only computed the volume of solids of revolution obtained by rotating around the $x$-axis the region between the graph of function and the $x$-axis. I made that choice on purpose. In this question we are asking students about a more general problem: rotate the region between two curves around various different axis.

Once again, we want students to understand how we set up volumes as integrals, rather than memorize formulas. The purpose of this question is not to give students formulas for a few more variations, but to have them derive them.

I have chosen an example where the functions are simple, the endpoints are simple, and the algebra is simple, so we can focus on what matters.

You could use this questions twice: once after Video 10.1 (to solve all the question with the “washer method”) and once after Video 10.2 (to solve all the questions with the “cylindrical shell” method).

I introduced this method with only one example in Video 10.2. This question simply asks students to write the general formula for an arbitrary function.

Students find this easy — assuming they have watched the video.

This is a good activity to finish the unit. It uses all the ideas from both methods but pushes students by being a bit different. Beware: this activity will take a very long time.

The two main difficulties are:

• The region is not described as being between the graph of two explicit functions.

• The axis of rotation is not one of the cartesian axes.

Still, if they fully understand the methods and why they work, they should be able, with enough time, to complete the tasks.

This question is probably too hard as a class activity. I suspect most students will simply say “I do not know what to do” and will wait without trying to do anything.

I save this question in case students are on top of everything, are finding the lesson too easy, and need a challenge. I have never needed to use it, though.

Solution: If the axis of the cylinders are the $x$-axis and the $y$-axis, cut the solid with planes parallel to the $xy$-plane (or perpendicular to the $z$-axis, if you prefer). The resulting cross sections will be squares.