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Here is the construction of the solid. Take an ellipse, make a copy of it, and put it on top of the original ellipse. Now turn the top ellipse by $90^\circ$ (quarter turn). Glue the two boundaries. I would like the height, volume, and the equation describing the ridge/boundary of this solid in terms of the major and minor axes.

Edit: Instead of two ellipses if you took a rectangle with sides $a$ and $b$ with $ b < a$ and followed the same procedure, you end up with box base a $b \times b$ square and of height $a -b.$ I wish I could take pictures of both solids I made and post it here. Hopefully this helps.

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    Your description is far too vague. Put the copy where? Turn about what axis? Glue what where?2012-02-20
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    Do you mean just two ellipses like this: $\{(x,y,z): x^2 + (y+1)^2 \leq 1, x^2 + (z-1)^2 \leq 1\}$? Or is it a 3D solid with a 2D ellipse (the first one) added? For the second case, you can easily find the volume with the Pappus-Guldin theorem (http://planetmath.org/encyclopedia/GuldinsTheorem.html).2012-02-20
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    @savick01: i don't think the solid i described is a solid of revolution.2012-02-20
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    If you say that you turn sth, there are two cases: you change its position (then it is similar to the set that I described) or you produce a solid of revolution. Here it is quarter of the full revolution, but we can divide by 4 (the theorem is true for any angle). Well, there is the third case if you turn around an axis intersecting the ellipse.2012-02-20
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    Could you elaborate a bit in your question to remove the ambiguities?2012-02-20
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    @savick0: i edited the question to make it clearer.2012-02-21
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    I believe the question is this: Cut two identical ellipses out of paper. If you put them on top of each other and glue the boundaries, you get a flat shape of zero volume. Instead, rotate one ellipse 90 degrees in the same plane. The shapes have the same perimeter, so it is still possible to make their boundaries coincide by bending them out of the plane towards each other. Gluing them together now produces a solid bounded by two ruled surfaces, which is what the asker wants to know about.2012-02-21
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    Aha, here's where I saw this kind of construction first: [they're called D-forms](http://paulbourke.net/geometry/dform/), and they're quite pleasing to the eye.2012-02-21
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    @narain: thank you for naming/classifying the solid.2012-02-21

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