BrainDen.com - Brain Teasers
• 0

# 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.

## Recommended Posts

• 0

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

##### Share on other sites

• 0

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!

##### Share on other sites

• 0

Is the three cylinder problem also doable without using calculus?

##### Share on other sites

• 0

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.

## Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

×   Pasted as rich text.   Paste as plain text instead

Only 75 emoji are allowed.

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.