It was my hope that a solution involving geometry, trigonometry and algebra would provide a solution.
(See my discourse below the horizontal line.)
As I analysed the object I kept finding shapes whose volume could not be described in simple lineal terms.
Please play with my thoughts below; I think that a Calculus result however was probably the only real outcome.
Steinmetz solids of which I had no concept were not mentioned in my previous proposal. Had they been, then this exercise would have been meaningless. Well, it did provoke some thought .. so .. good.
Cee-Anne, through Larry, brought to me the link that Richard kindly offered. It is very informative and well worth the study. I can see some of my thoughts being reflected within some of those images ( er… look down to 3 cylinder intersections).
I like Joel's clarity in a concise development of his solution; 2-sqrt(2)! Note! This agrees with the linked article. [I must study those integrals, thanks Joel].
Why Pi does not appear in this answer is a question I cannot answer.
I suggest that anyone wanting to explore this further might create/add to a 'Steinmetz' thread in Forums/General Discussion.
At best, I managed to see the curved areas as triangular curved sheets, or rather prisms.
_______ _______
|\A /| |\A| /|
|B\ / | |B\|/ |
| / \ | | /|\ |
|/ \| |/ | \|
[Upright cylinder - diameter and height are equal]
I could cut an 'upright' cylinder into 1/4's, in the x,z plane; first diagram.
It takes some thinking, but the volume of A is equal to B.
I could make it into 1/8's with a cut down the z plane to the y axis.
This is good, because I can calculate 1/8 of a cylinder's volume.
But .... as I move my focus along the diagonal separating the letters A and B, the square which we may now consider to represent the cylindrical body diminishes in size; in fact this square actually represents a 3D segment of the face of the cylinder whose volume reduces with the angle of the chord.
Suddenly I notice that the volume of A is greater than B to the point when A=B=0!!
The volume of these pieces is clearly a function of the chord angle. As that angle decreases from 180° the length of the chord also decreases (and thus the area of the square) and the height of the segment. For this problem the chord subtend angle is 90°.
Does a non-calculus solution exist?
May I reiterate:
I suggest that anyone wanting to explore this further might create/add to a 'Steinmetz' thread in Forums/General Discussion.
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