# Volumes of Infinite Collection of Cubes Essay

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This essay continues the scenario of Essay 1. You will now describe some further counter-intuitive properties of infinite sets for an audience that has had exposure to calculus II.^{[if !supportFootnotes][1][endif]} Think of this essay as an article that might appear in a newsletter for undergrads interested in mathematics and science that have studied integration and infinite series, but perhaps need some reminders about those topics.

Recall that your essay should have a title, introduction and summary, as well as a main body.

The Blocks Unlimited store has expanded its offerings to include an Ultra Deluxe Set of blocks. In this infinite set the n^{th} block has side length.

Show that this set contains the Deluxe Set as a subset (which in turn, as you should have noted in Essay 1, contains the Starter Set as a subset). Thus argue that the Ultra Deluxe Set, when stacked, would reach infinitely high.

Recall the special paint from Essay 1. Show that this set cannot be painted with a finite amount of the paint. Show that, in contrast, a finite amount of ordinary paint (in fact less than 3 cubic feet) would suffice to fill the blocks if they were hollow.

Compare the Ultra Deluxe Set with the infinite horn generated by revolving the graph of from to about the x-axis.

Show that this horn has finite volume but infinite surface area and so is somewhat like the Ultra Deluxe Set. Give the value of the volume.

The examples given in the previous paragraph are to be discussed in detail in Essay 2

You should include in your essay some additional examples of your own creation or a discussion of the following:

Consider the horn generated by revolving the graph of[ to about the x-axis. Is the volume finite or infinite? What about the surface area?

This assignment is a sequel to earlier assignments of Baldwin, Berman, Thulin and Radford and is based eventually

on an exercise in Writing in the teaching and learning of mathematics, J. Meier and T. Rishel, MAA 1998.

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