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Three intersecting cylinders


Question

A classic problem solved by Archimedes before the invention of calculus, concerns two circular cylinders intersecting at right angles.

If each cylinder has a radius of 1 unit, what is the volume of space that is common to the cylinders?

For warm-up, you can figure out the answer - using calculus if you like.

Here's the puzzle of the day:

What is the volume of space that is common to three orthogonally intersecting cylinders of unit radius?

In both problems the axes of the cylinders intersect.

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I've done the 2 cyclinder problem before...so that really helps with starting this problem. I'll leave that one to someone else...

I beleive this can be represented by this integral (this could be very wrong, as it's been a while...):

16 * 0pi/4 01 0√(1 - r^2cos^2Θ) r dz dr dΘ

So, solve the inner integral:

16 * 0pi/4 01 r * √(1 - r2cos2Θ)dr dΘ

Then the next integral (this is the one I'm really not sure I did right):

-16/3 * 0pi/4 ((sin3Θ - 1) / cos2Θ) dΘ

Finally, the last integral:

16 - 8√2...or approx 4.69

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I've done the 2 cyclinder problem before...so that really helps with starting this problem. I'll leave that one to someone else...

I beleive this can be represented by this integral (this could be very wrong, as it's been a while...):

16 * 0pi/4 01 0√(1 - r^2cos^2Θ) r dz dr dΘ

So, solve the inner integral:

16 * 0pi/4 01 r * √(1 - r2cos2Θ)dr dΘ

Then the next integral (this is the one I'm really not sure I did right):

-16/3 * 0pi/4 ((sin3Θ - 1) / cos2Θ) dΘ

Finally, the last integral:

16 - 8√2...or approx 4.69

Bravo, for accuracy and beautiful symbols. Two coveted stars!

post-1048-0-53237200-1344278051.gifpost-1048-0-53237200-1344278051.gif

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