Diverging area under $\frac{1}{x}$ vs. Gabriel's Horn's finite volume I am currently studying Calc 2 in high school, and we recently talked about improper integrals, and as an example, our teacher talked to us about Gabriel's Horn, where the area represented by $\int_{1}^\infty \frac{1}{x} dx$ is revolved around the x-axis. 
If $\int_{1}^\infty \frac{1}{x} dx$ diverges, then it could be said that the area between the function $\frac{1}{x}$ and the x-axis from $x=1$ to $\infty$ is infinite. 
My question is, how come when this "infinite" area is revolved around the x-axis, it results in a finite amount of volume? I understand that once you set up the equation to find the volume you get this:
$$2\pi\int_{1}^\infty \Bigl(\frac{1}{x}\Bigr)^2 dx$$
and that $\int_{1}^\infty \bigl(\frac{1}{x}\bigr)^2 dx$ converges to $\pi$, therefore the volume is finite. I just can't wrap my mind around the fact that you can get a finite amount of volume from a seemingly infinite amount of area. 
Note: by area I refer to the area between the function $\frac{1}{x}$ and the x-axis from $x=1$ to $\infty$, not the surface area of Gabriel's horn. 
 A: Take some Play-Doh and roll it into a ball. It has some volume and some surface area. Now roll it in one direction until it extrudes into a cylinder. The volume is unchanged but the surface area has increased.
Let's call the cylinder volume $V$, its surface area $A$, its length $L$, and roll it some more until the thickness of the cylinder is half of what it was. When the thickness is halved, the cross-sectional area decreases by a factor of $2^2=4$, since the area of a circle is $\pi r^2$. All this volume has to go somewhere, and it goes into the length. The length of our cylinder is now four times what it had been. So we have $L' = 4L$, $A' = 2A$, and $V'=V$ where the prime denotes the one that we made thinner. We can also add $r' = \frac{1}{2}r$ to this list if we like.
Notice that things are changing at different rates. The length changes the fastest, the surface area and radius changes at the same (slower) rate, and the volume doesn't change at all. This is connected to the 2-dimensional version of the cube-square law, which you may be familiar with from science classes. What we are doing here is sorta the same thing in reverse. We are holding the volume fixed, but increasing the surface area and the length. Hopefully it seems relatively intuitive that you can have increasingly larger surface areas with a fixed volume for the cylinders.
Gabriel's horn is the exact same thing, done a little cleverly so that instead of getting arbitrarily large surface area you actually get infinite surface area.
A: The area closer to the x-axis contributes less volume. For example,  If you take a rectangle and revolve it around a line, you get a lot less volume when the line you’re revolving around is really close. Since the area here is all really close to the x-axis, it contributes very little volume. 
This intuition doesn’t tell us whether each rectangle will make a small enough contribution for the overall integral to converge, but when we return to analytic methods we find that it does.
