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I'm confused by these "surfaces". I know how to find volume when curve rotates around $Ox$ or $Oy$ line, but I don't know what to do with surfaces. I need to find volume bounded by surfaces $$ \left(x \over a\right)^{2} + \left(y \over b\right)^{2} = 1\qquad\mbox{and}\qquad \left(x \over a\right)^{2} + \left(z \over c\right)^{2} =1\,,\qquad\qquad a,\,b,\,c\ >\ 0 $$ Also, I'm not allowed to use multiple integrals. So, could someone provide me a hint, or maybe if there is any formula.

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  • $\begingroup$ While this question is unsolved for general elliptical axes lengths, I'd like to make a link to the related post. Also see the meta post. $\endgroup$ – Lee David Chung Lin Jan 22 at 11:49
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You're being asked to compute the volume of the intersection of two cylinders, apparently called the Steinmetz solid. The Wikipedia entry gives the answer $16r^3/3$ when $a=b=c=r$ and shows how to get it without doing multiple integrals. A simple scaling argument gives the answer $16abc/3$ for your problem.

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