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Confusion - what confusion might you be referring to? ;-)
Any mathematically defined volume that is 'under water' (perhaps also
in contact with the bottom my aquarium) must experience a NET upward
force that is equal to the weight of the displaced water.
Sounds simple enough to me.
uh, I was thinking that the total SIX sided integral would have to be
the same for the same volume.
(regardless of what was filling up that volume.)
I guess I AM still confused.
I was thinking that he mathematical integral didn't depend on silly
details like what was contained within the volume. Or whether it was
a micron or so above the glass.
Straighten me out here please.
... Just a thought that might help. Imagine that you place the physical
cube JUST in contact with the bottom so that all fluid is excluded,
but the bottom exerts NO significant force on the cube. Hopefully
you see that the cube will be DRIVEN downward into MORE significant
contact with the bottom. It is the resulting compression of the
bottom that results in it exerting a greater--and ultimately
balancing--upward force. The same mechanism explains how horizontal
surfaces always manage to exert JUST the right amount of upward
force on whatever sits on them.