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!