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The analogy for cross-sections is easy since we can think of how slices of bread can make up a loaf.

But what would be the analogy for cylindrical shells?

Regarding shapes, apparently there's something called a "bundt" cake, but I suppose a bagel or donut (aka torus) would be more well known. However, I'm not sure if idea of gradually building up the solid by cylinders works?

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    $\begingroup$ How about these toy cups? At least they're concentric shells (if you ignore the lip of each cup). images.app.goo.gl/YCZjKDRsydtDtYHn7 $\endgroup$ Sep 9 '20 at 4:45
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    $\begingroup$ The arts and crafts type students will know about spinning pottery wheels, and at least in the 1990s in the U.S., pretty much every student knew about the spinning pottery scene in the 1990 movie "Ghost". However, the Russian dolls that @mathkb8 brought up is a much better analogy, since in the case of spinning pottery you're not really building up a volume by shells. But many students will probably not know what Russian dolls are unless you have a picture. $\endgroup$ Sep 9 '20 at 18:10
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I usually reference how we are filling the volume with nested cans (they can all visualize metal food cans with the top and bottom removed) and that works pretty well. Usually when I mention this, a student will ask if cylindrical shells are like Russian nesting dolls - which I think is a good analogy as well, but not all students may know of these dolls.

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  • $\begingroup$ I use this image (nested soup cans) and also the idea of a cookie cutter type device making those cuts. $\endgroup$
    – Sue VanHattum
    Sep 9 '20 at 4:09
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    $\begingroup$ I like the cookie cutter idea! The nested cans or Russian dolls seems to really work - I have two sections of calc 2 this semester and they seem to overwhelmingly prefer cylindrical shells for volume problems (I teach both shells and washers/disks, but then let them choose how to approach the problems on quizzes/exams rather than forcing a particular method). $\endgroup$
    – mathkb8
    Sep 9 '20 at 23:00
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How about telescopic collapsible camping cups?


      

Image from wish.com.


Similar to @BrendanSullivan's suggestion.
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