How can a structure have infinite length and infinite surface area, but have finite volume?

Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis.

One Dimension - Clearly this structure is infinitely long.

Two Dimensions - Surface Area = $2\pi\int_∞^1\frac{1}{x}dx = 2\pi(\ln ∞ - \ln 1) = ∞$

Three Dimensions - Volume = $\pi\int_∞^1{x}^{-2}dx = \pi(-\frac{1}{∞} + \frac{1}{1}) = \pi$

So this structure has infinite length and infinite surface area. However it has finite volume, which just does not make sense.

Even more interesting, the "walls" of this structure are infinitely thin. Since the volume is finite, we could fill this structure with a finite amount of paint. To fill the structure the paint would need to cover the complete surface area of the inside of this structure. Since the "walls" are infinitely thin, why would a finite amount of paint not be able to cover the outside of the "walls" too?

Please help me make sense of this whole thing.

Solution 1:

It seems you already have made sense of this whole thing from a mathematical point of view (it has infinite surface area and finite volume -- there's no contradiction here). The paradox is that infinity, at times, does not match well with our day-to-day experiences.

Here's a related example (and hopefully a little bit easier to picture): Take a unit square. Cut it in half vertically down the middle -- forming two pieces. The area clearly remains the same, but the total perimeter (counting both pieces) has increased by 2. Keep cutting the right-most piece vertically down the middle forever. The area remains unchanged throughout, whereas the perimeter is $4+\lim_{n \rightarrow \infty} \sum_{i=1}^n 2=\infty$.

Now, if you wanted to make it look more like a horn, you could move the pieces around, for example:

which still has area 1 and perimeter infinity. If you wanted to make it have volume 1 and surface area infinity, replace "square" with "cube".

Solution 2:

The mathematics has already been explained; when the mathematics contradicts your intuition the answer is to ignore your old intuition and find new intuition.

As for the physics, Gabriel's horn is in no meaningful sense a physically realistic object. One way to summarize the lesson here is that the assumption that very thin objects have zero thickness is only reasonable if this causes a reasonable amount of error in one's computations, but in the construction of Gabriel's horn the thickness of the membrane comes to dominate the computation and it is no sense reasonable to ignore this error. Any time a mathematical model fails to accurately reflect reality one should question the assumptions that go into building that model, and to the extent that this is a mathematical model of anything, the assumption that thickness is negligible is the one that should be thrown out. If you try to repeat the construction of Gabriel's horn with a fixed finite thickness you will, of course, find that the resulting object has infinite volume.