Can anyone help me figure out how to find the shared volume of two rotations of a line, one around the x axis and one around the y? I can't find anything online on the subject, I think the regular integration methods drown out my search attempts.
To keep it simple, say we have a curve $\frac{x^2}{8}$, bound between x=2 and x=8, shown below.
Revolve it around the x-axis and get a trumpet-like shape, and calculate the volume. Then, revolve it around the y-axis to get maybe a bathroom sink shaped object, I can calculate that too. Now I need to know how to calculate the volume of their shared space, so I can apply it to my problems.
Apparently, it's not that hard (U.K. A level standard) but all the curved edges following curved edges are freaking me out!
Any direction would be appreciated!
W
From your description I believe you are rotating only the curve itself, as defined for $2\le x\le 8$, and not the area under the curve shaded in your graphic. That is the straightforward conclusion from your words and one aspect of your graphic (the bounds on $x$)--if that understanding is wrong, explain in detail just what you mean.
Given your two state rotations, the only intersection between those rotated volumes is the curve itself, which has volume zero. Therefore, the shared volume is zero.
To see the intersection, when rotating around the $x$-axis you use the area below the curve down to the $x$-axis. When rotating around the $y$-axis you use the area above and to the left of the curve.