Does Simplifying Make Things Simpler?

I taught a class today on volumes of solid of revolution; specifically the ones with holes through them by the so-called “washer method.”  For this method you often see the formula

\displaystyle V=\pi \int_{a}^{b}{{{R}^{2}}-{{r}^{2}}}dx

Where R is “outside radius” and r is the “inside radius” and both are functions of x.

It seems to me that this “simplified” form is unduly complicated.

We first worked a problem where a region was rotated around its horizontal edge (Disk method). The region was between y=\sqrt{x}, and the line y=-3 between x = 0 and x = 4, revolved around y=-3.

\displaystyle V=\int_{0}^{4}{\pi {{\left( \sqrt{x}-\left( -3 \right) \right)}^{2}}dx}.

Then I asked them to change the region to that between the graph of y=\sqrt{x} and the line y=\tfrac{1}{2}x again revolved around y=-3. These graphs intersect at x = 0 and x = 4. (How convenient!).

Someone immediately had the idea to revolve the line only and subtract the answer from the last answer:

 \displaystyle V=\int_{0}^{4}{\pi {{\left( \sqrt{x}-\left( -3 \right) \right)}^{2}}dx}-\int_{0}^{4}{\pi {{\left( \tfrac{1}{2}x-\left( -3 \right) \right)}^{2}}dx}

Done!

Isn’t that good enough? Is there any need, ever, to set up the washer? Can’t you always subtract the inside volume from the outside volume?

Now I know that

\displaystyle \int_{0}^{4}{\pi {{\left( \sqrt{x}-\left( -3 \right) \right)}^{2}}dx}-\int_{0}^{4}{\pi {{\left( \tfrac{1}{2}x-\left( -3 \right) \right)}^{2}}dx}=

\displaystyle \pi \int_{0}^{4}{{{\left( \sqrt{x}-\left( -3 \right) \right)}^{2}}-}{{\left( \tfrac{1}{2}x-\left( -3 \right) \right)}^{2}}dx

And the latter is shorter and simpler to look at – only one \pi and only one integral sign, but which is really easier to understand and set up? Which shows you really what you’re doing?

Just sayin’.

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4 thoughts on “Does Simplifying Make Things Simpler?

    • Certainly. Why not? Any correct mathematics that solves the problem will get credit on the AP exam. Readers do not take off for correct mathematics.
      As the previous comment by Brianne Allbee points out, using separate integrals actually helps students understand the situation and makes it easier to get the correct answer.

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  1. I love this. I started showing my students to use separate integrals a couple of years ago and their ability to solve washer problems dramatically increased. I have found my students can visualize the situation much easier this way. We do discuss how it could be written as one integral as that is how the scoring guideline for free response problems are written, but most of my students stick with the separate integrals.

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