Tuesday, August 31st, 2010...1:38 pm

A Yoda Revolution

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This semester, I’m teaching Calculus II. Yesterday, students submitted homework on finding volumes of solids of revolution. As a quick primer on these shapes, a solid of revolution is generated by rotating a planar region about an axis. For instance, rotating the right triangle below about the vertical axis generates the cone to the right.

For fun, we started class, having completed an entire homework set on solids of revolution, by creating a fanciful solid of revolution. Rather than using a planar region, suppose we take a 3D solid itself and rotate it. What will happen? Let’s try. In fact, let’s just do it. Do or do not….there is no try! If you recognize that quote, you may realize we are about to rotate a 3D image of Yoda as seen below.

Take a moment and imagine rotating this image of Yoda about the vertical axis. What will you get? Can you see it? It takes a couple of minutes with the math software I use in class but homework is returned as we watch the shape unfold. And here it is:

We then rotated it and several students commented that this angle of the Yoda solid looked like it should be attached to the wing of a commercial jet!

What’s the volume? In terms of the tools we had in class, we’d need to find a planar region that would produce such a shape and then describe that planar region as a function! If you are already scribbling away at such calculations, may the force be with you.

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