Shell approximation of solid of revolution II

by mathgrrl, published

Shell approximation of solid of revolution II by mathgrrl Apr 22, 2014


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The model pictured on the left is the solid obtained by revolving the region between the graph of y=(x-1)(x-3)^2 and the x-axis on [1,3] around the y-axis. The model in the back is an approximation of that solid using 8 shells, and the model on the right is an approximation with 16 shells.

This particular volume is the same one computed with a definite integral at https://www.khanacademy.org/math/integral-calculus/solid_revolution_topic/shell-method/v/shell-method-for-rotating-around-vertical-line, in case you feel like doing some calculus.

For more information about the math behind approximating volumes with shells, see the link below.

MakerHome, Day 238:


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I am trying to lead my Calc students through a process like this. It's like the blind leading the blind, though. Would you be willing to share the exact steps? I've been getting stuck with Mathematica images that I don't know how to export to the right file for our local 3D printer.

I made the model in OpenSCAD and you can read the steps at this link: http://makerhome.blogspot.com/2014/04/day-238-shellmaker-2.html

You can see an earlier version here: http://makerhome.blogspot.com/2014_04_20_archive.html

And en even earlier version made with Tinkercad here: http://makerhome.blogspot.com/2014/01/day-143-volumes-of-hanoi.html

For a class I think the easiest thing to do would be to import a 2D image of the curve you want to rotate into Tinkercad and then use it as a guide to create shells or slices for approximating the rotated volume. Or, if your students like code, they could modify the OpenSCAD design for different curves and numbers of shells.

Good luck, I hope it goes well!

thanks! I was looking for exactly this project for my math class! Can you tell me your printer settings and how long it took to print?

I dig your designs... These could make for great conceptualization learning tools for advanced integration techniques.