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TomasB Inner circle Sweden 1144 Posts |
Imagine the function f(x)=1/x defined between 1 and infinity. Imagine it rotated around the x-axis to create a funnel with an opening with a diameter of 2.
What is the volume of this funnel? What is the area of this funnel? /Tomas |
Daegs Inner circle USA 4291 Posts |
Oooh I love this one.
I really liked the proposal that our universe was shaped in this manor, which would give both the finite camp and infinite camp good answers and I think might explain some things... |
Nir Dahan Inner circle Munich, Germany 1390 Posts |
Tomas,
I assume no calculus is allowed... Nir |
TomasB Inner circle Sweden 1144 Posts |
Nir, solve it by any means you like. It's the result in this case that is interesting, not how you get there. As long as you do it right of course.
/Tomas |
leonard Regular user North Carolina 148 Posts |
Tomas,
I remember seeing a cartoon about this in High School. I believe it was called "Infinite Acres," and dealt with a character building a house as you describe. Seems he could never store enough paint to get the job done. Mel Henrickson's name comes to mind (one of my math profs in college), although I think he had something to do with the cartoon presentation and not the problem itself. I will give this a go later this week. leonard P.S. I look forward to Nir's non-calculus solution. |
NCR New user Northfield Mount Hermon, MA 61 Posts |
Wow. I don't even know where to start...And I thought I was at least not bad at math...
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TomasB Inner circle Sweden 1144 Posts |
NCR,
Big spoiler below... . . . . . . . . . . . . . . . . . . . . . If you cut a disc at point x out of the funnel it will have the area pi*f(x)^2 and the circumferance 2pi*f(x). Multiply those with an infinitisemal thickness (dx) and you will get a local volume respective a local surface area. Sum those up from 1 to infinity and you will find the answers. /Tomas |
TomasB Inner circle Sweden 1144 Posts |
Hmmm, I'll just write a procedure to get the answer, but please ponder what the answer actually means. I've called the natural logarithm "ln".
V = integrate from 1 to inf(pi*f(x)^2 dx) = pi*integrate from 1 to inf(1/x^2 dx) = pi*[-1/x]:1:inf = pi*(-1/inf + 1/1) = pi A = integrate from 1 to inf(2pi*f(x) dx) = 2pi*integrate from 1 to inf(1/x dx) = 2pi*[ln(x)]:1:inf = 2pi*(ln(inf) - ln(1)) = inf In other words you have an object which you can't paint, since its surface is infinite, but you CAN fill it with paint. /Tomas |
Nir Dahan Inner circle Munich, Germany 1390 Posts |
Tomas,
I think there are other interesting examples of surfaces you can paint (area is limited) but has infinite circumference (you can't trace it with a pen). I believe the koch fractal is one of them but I may be wrong. Nir |
TomasB Inner circle Sweden 1144 Posts |
Nir, you are very right. A fractal border can be infinite but hold a finite area. I _think_ you can find borders that are fractal but still finite though, so they should be able to trace with a finite amount of ink but since you have to change direction of the pen an infinite number of times to draw it...
/Tomas |
Nir Dahan Inner circle Munich, Germany 1390 Posts |
Tomas,
along these lines, I remember years ago in my first year in uni the lecturer showed us a function which is continuous but can not be derived at any point. Nir |
TomasB Inner circle Sweden 1144 Posts |
Nir, there are many examples of such functions. Usually an infinite sum of sawtooth shapes that gets infinitely denser. Quite similar to the Koch fractal you wrote of, actually.
/Tomas |
2hot New user Sydney 89 Posts |
Dude you didn't even give us the boundary for where the funnel ends
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TomasB Inner circle Sweden 1144 Posts |
Ok, 2hot, you may give the answer where the boundary is [1,t] where t is a real number and t > 1.
/Tomas |
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