bonanova 85 Posted August 6, 2012 Report Share Posted August 6, 2012 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. Quote Link to post Share on other sites

0 Pickett 13 Posted August 6, 2012 Report Share Posted August 6, 2012 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 * _{0}∫^{pi/4} _{0}∫^{1} _{0}∫^{√(1 - r^2cos^2}^{Θ) }r dz dr dΘ So, solve the inner integral: 16 * _{0}∫^{pi/4} _{0}∫^{1} r * √(1 - r^{2}cos^{2}Θ)dr dΘ Then the next integral (this is the one I'm really not sure I did right): -16/3 * _{0}∫^{pi/4} ((sin^{3}Θ - 1) / cos^{2}Θ) dΘ Finally, the last integral: 16 - 8√2...or approx 4.69 Quote Link to post Share on other sites

0 bonanova 85 Posted August 6, 2012 Author Report Share Posted August 6, 2012 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 * _{0}∫^{pi/4} _{0}∫^{1} _{0}∫^{√(1 - r^2cos^2}^{Θ) }r dz dr dΘ So, solve the inner integral: 16 * _{0}∫^{pi/4} _{0}∫^{1} r * √(1 - r^{2}cos^{2}Θ)dr dΘ Then the next integral (this is the one I'm really not sure I did right): -16/3 * _{0}∫^{pi/4} ((sin^{3}Θ - 1) / cos^{2}Θ) dΘ Finally, the last integral: 16 - 8√2...or approx 4.69 Bravo, for accuracy and beautiful symbols. Two coveted stars! Quote Link to post Share on other sites

0 kbrdsk 2 Posted August 12, 2012 Report Share Posted August 12, 2012 Is the three cylinder problem also doable without using calculus? Quote Link to post Share on other sites

0 bonanova 85 Posted August 13, 2012 Author Report Share Posted August 13, 2012 Is the three cylinder problem also doable without using calculus? Well to be fair, Cavalieri's principle was the calculus dodge for the two cylinder calculation. It's a near cousin to the integral calculus. The three cylinder problem might succumb in a similar way. Quote Link to post Share on other sites

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## bonanova 85

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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