Well, it's Gabriel, Gabriel playin'!
Gabriel, Gabriel sayin'
"Will you be ready to go
When I blow my horn?"
- Cole Porter, Anything Goes
The commenters in last week's Sunday Function proposed an excellent idea for this week. As we did then, we'll start simple and work up to it. Graph the curve f(x) = 1/x.
Now take the horizontal axis and think of it as an axle, one that can rotate smoothly. Hook a motor onto it and spin it up to a nice fast clip. The graph of the function will trace out a surface in three-dimensional space. In the business, we call it a surface of revolution. It looks like this:
Now I've truncated the plot at x = 4 for reasons of space. But in reality you can imagine that in fact the surface keeps going on forever in that direction in the shape of an ever-narrowing trumpet. That shape is called Gabriel's Horn, and though I'm leaving off the actual explicit specification for the expression describing it, it's still a function (of two parametric parameters or of y and z) nonetheless.
There's two things we might want to calculate. First, the volume contained in the trumpet. Imagine you have a penny, and you drop it into the trumpet such that it acts as a plug or a manhole cover. It's perfectly perpendicular to the x-axis. The penny will have a radius of f(c) = 1/c, because at the particular point c that it sticks it clearly must have the same radius as our original function f(x) evaluated at c. What's the volume of that penny? It's the area, pi*r^2, times the thickness of the penny. Knowing that r = 1/x for whatever generic x the penny sticks at and calling the thickness dx, we have a volume (call it dV) for the penny of dV = pi/x^2 dx. So if we fill the entire trumpet with custom pennies of appropriately varying radii and add up their volumes, we'll have the volume of the trumpet. Looks like a job for calculus!
So despite being infinitely long, the trumpet has a finite volume. It's a little counterintuitive, but it's perfectly true nonetheless.
What about the surface area? It's a little harder to find the expression for surface area of a solid of revolution, but you can take my word that it's this, with the third expression having our particular function plugged in:
That integral looks fairly tricky to evaluate directly. I think it could probably be done, but we won't have to. The term under the square root is bigger than 1, and so the 1/x term is a lower limit for the behavior of the integrand. But 1/x diverges, so the integral is infinite. Our infinitely long horn has infinite surface area.
Infinite surface area and finite volume. That's weird. As people like to say about this shape, you can fill it with paint, but you can't paint it. It's true, but it's not so implausible as it sounds because physical paint isn't just a pure 2d surface. Real paint has thickness. If you poured real paint into a real Gabriel's Trumpet, eventually it would fail to keep coating the interior once it narrowed down to a thinner diameter than a paint molecule.
Even Gabriel might think that's pretty cool.
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That's coolio that the surface area diverges while the volume converges.
It's actually not too hard to evaluate the SA integral directly. Clear the denominators in the square root to get (1/x^3)sqrt(x^4+1), use integration by parts with u=sqrt(x^4+1), dv=1/x^3, and the resulting integral is a simple inverse trig substitution. You end up with indefinite integral = -sqrt(x^4+1)/(2x^2)+arcsin(x^2)/2.
I believe that should be an arcsinh in your last term, but other than that you're right. Thanks!
Very interesting, thanks!
Your example of painting the surface could be expanded. For any finite thickness of paint on the outside of the trumpet, the volume of paint would be infinite. The inside of the trumpet is at distance 1/x from the axis. The outside of the paint lying on the trumpet would be described by a different equation, (1/x)+c. Hence, paint volume infinite. If we painted with more sensitivity to context, so that there would always be paint, but not overpoweringly so, the outside of the paint might be described by c/x, c slightly greater than 1, then the volume of paint would be finite (the difference between the volumes of two Gabriel's horns). We could also model the outside of the paint by 1/(x-c), c slightly greater than zero, which would use a different volume of paint.
As you say, paint molecules are of finite size, but, given that Gabriel's Horn is a Godly or mathematical object, we might use equally mathematical paint.
Deeper, deeper...
http://en.wikipedia.org/wiki/Pseudosphere
I thought gabriel's trumpet was that hypersphere thingo - the tractrix. I suppose anything that's kind of trumpetlike manages to acquire the name.
Speaking of which - how do you calculate the curvature of a surface?
Your first radical sign has a 90 degree angle, the second has is greater than 90 degrees. Oh, the OCD!
WELL....ALL I CAN SAY IS IT WAS JUST A RESULT OF CALCULATIONS IT MAYBE SOMEHOW HAVE DIFERRENCES WITH NATURAL QUALITATIVE ANALYSIS''''
Another fun fact: the trumpet is the same (IIRC) as the pseudosphere, which means that it has constant negative Gaussian curvature along its surface.
It is a convoluted way to look at it, but if you want to paint the outside of the trumpet you need an infinite number of cans of paint; if you want to paint the inside of the trumpet, the paint will be part of the inside volume, so you need only a finite volume of paint.
The trumpet is quite extraordinary really. You will see an infinite number of cans of paint on the outside. The volume of paint is quite finite in this regard.