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Recently I've been thinking about solids of revlution, and thought about an interesting experiment. Can you rotate functions around, for example, the line $f(x)=x$? And consequently, could you rotate a function around another entirely different function?

To rotate solids around the line $f(x)=x$, I thought you might have to make the "cylinders" go diagonally, but I can't think of how you would do that (distance formula?). I am thinking that there is probably an easier technique, perhaps using the "solids with known cross sections" method.

About the second part of my question, imagine "rotating" the rectangle formed by $\int_{-1}^12dx$ around the parabola $g(x)=-\frac12x^{2}-1$. Does this idea make any sense? I would imagine the result would be very convoluted and weird (maybe relates to linear transformations).

Thanks!

Tdonut
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    Yes, one may form rotations about any axis - there is nothing special about one coordinate system or another. But, how do you intend to define rotations about a non-linear axis? What is the transformation? – Mark Viola Apr 14 '15 at 16:38
  • @Dr.MV, I am not sure how I would define rotations about a non-linear axis. Perhaps a (very weird) möbius transformation of some type? – Tdonut Apr 14 '15 at 16:44
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    You can use some linear algebra (rotation matrices) or Pappus's Theorem to do this. However, it's a really cool exercise to use cross-sections to find the volume when you rotate the region bounded by $y=x$ and $y=x^2$ about the line $y=x$. You have to be very careful to keep track of your variables. – Ted Shifrin Apr 14 '15 at 16:45
  • Hi, not to seem like “spam”, but I think I have at least found some insight into your question. You can first have a point on the relation and/or functional axis and then draw its normal line in the direction of the other relation/function. Then make the line go in the other direction and be the same length from the point to the other function. Please visit my question to see this simplified! Part of Solution – Тyma Gaidash Apr 02 '21 at 21:29

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