So there's very similar other effects about differences in dimension, the difference between, say, surface area and volume, and I want to talk about some interesting mathematical examples now. And one of those examples is the paradox of Gabriel's horn. So Gabriel's horn, so we, we look at the function of one over X. So this is Y equal one over X. And we start at the, uh, point one.
We're just gonna look at this part of the curve here. So it goes all the way out to infinity. So then to form Gabriel's horn, we take that line, and we revolve it around the X-axis. So we make a kind of, um, symmetric kind of shape like this. Yeah.
And it's kind of like a horn from heaven. Maybe we can imagine it with a sonorous, uh, tone, maybe this multi tone coming, uh, from heaven, and this is why it's called Gabriel's horn. Yeah. So Gabriel's horn is obtained by revolving this function around the X-axis and making this kind of surface. So I could put my arm in it sort of like a, a horn.
Okay. So now the paradox of Gabriel's horn is the following observation. Well, uh, I want to ask, what is the volume of Gabriel's horn? Yes? I mean, it's an infinite object because it, it keeps going forever, but it's one over X, and so it's very thin out there when you go very far.
And so we can compute the volume of this. And in order to compute the volume of this, well, it's a kind of standard problem in calculus, um, to compute the volume of a, of a surface of revolution. Um, and the way that you do it is in the typical calculus manner. You imagine sort of slicing it into cross-sections. So we're gonna slice this horn into these kind of, uh, disks.
Yeah. I'm thinking of the solid, the, the volume of the solid, the part inside the horn as well. That's what I'm talking about. Um, and if I slice it like this, then the, uh, if I'm at point X here, the radius of that disk is one over X because that's the function that we're rotating. Yes.
And maybe the thickness, if you know how to do this, is DX, the infinitesimal DX. Yeah. So the, the volume of one disk is the area of that disk times the thickness, but the area is pi R squared. So therefore, the volume of one disk is pi times one over X squared DX. So that's the area times the thickness.
Yeah. So the total volume will be the integral from one to infinity of the volume of each disk. Yeah. So we took the whole thing. We slice it into these disks.
We calculate the volume of each disk. That's the integrand, and now we're gonna add them up. That's exactly what integration does. So this is equal to the integral from one to infinity of pi over X squared DX. Okay.
But I can compute this integral. This is a elementary calculus integral. The integral of one over X squared is minus one over X. So we have here minus pi over X from one to infinity. And when I put in infinity in for X, the limit, uh, will be zero.
So it's zero minus using the fundamental theorem, uh, the value when I put one in. Um, uh, but there's a minus sign here, so it's minus, minus pi over one, and so I get pi. So the volume of Gabriel's horn is precisely pi, which is a finite number. That's the first paradoxical part about Gabriel's horn. It's an infinite object, but it has a finite volume, pi.
So the volume of Gabriel's horn is finite. Okay. That's nice. But the second part of the paradox is to ask, what is the surface area of Gabriel's horn? Yes.
So now instead of asking the volume, I want to ask the surface area. Well, for the surface area, we don't want the volume of this disk, but rather we want to concentrate on this band on the outside, which if you think about it, is the what's called the frustum of a cone. It's slightly angled, right? And the angular piece here is, is commonly called DS, which is the square root of DX squared plus DY squared. And I can think of that as if I factor out a DX, one plus DY DX squared DX here.
So the length, the infinitesimal, uh, um, uh, length of, of the frustum part is this square root times DX. So the total surface area will be the integral from one to infinity of the surface of one frustum, but that's the width of that frustum times the circumference, and the circumference is, is pi times the diameter. So that would be two pi over X. So we have two pi over X times this square root thing, one plus DY DX squared DX. And one-- And DY DX squared, well, Y was one over X, um, uh, um, which is like X to the minus one, and so DY DX is, uh, is minus one over X squared.
And so this turns into one over X to the fourth. Okay. So this is more complicated, but one can calculate. In fact, we can just ignore this square root. It's always at least one.
So this is bigger than or equal to the integral from one to infinity of two pi over X- DX, if I just observe, this is at least one, so this whole thing is at least as big as this, and this is two pi log X from one to infinity, which is infinity. Okay. So the point I'm trying to make here is that the surface area of Gabriel's horn is infinite, but the volume of Gabriel's horn is finite. Okay. The surface area is infinite, and the volume is finite.
But how could that be? I mean, can't we just fill it with paint? Suppose I, I point it down, and I just fill it with paint. It has a finite volume, so with finitely many buckets of paint, I can fill it up, and that paint would be touching every part of the surface. So with a finite amount of paint, I can paint Gabriel's horn, right?
So that's the kind of puzzle of Gabriel's-- the paradox of Gabriel's horn is that it's a, it's a geometrical object that we can understand in a deep way, and yet it has finite volume and infinite surface area. Okay. And so what about this filling with paint idea? I mean, does it really work? Does it convince you if you have an object, you know, a container, and you fill it with paint, then would it ever take more paint than that to paint that surface?
And I would say, "Well, actually, we're cheating a little bit with that argument," because, um, the Gabriel's horn is getting thinner and thinner as you go out here, and so the paint that's inside Gabriel's horn was spread very thinly when you go very far out. If you say, "Well, to paint a surface, you know, there should be a uniform one millimeter thickness of paint on it," then eventually we, we weren't obeying that one millimeter thickness because the horn itself was less than one millimeter across. And so even though the horn was full of paint, it doesn't mean that we've painted to a uniform thickness. Yes. And so that's a way of seeing, well, look, just filling Gabriel's horn with paint shouldn't count as painting the surface because you've spread the paint so thin in the part that's way out there, but the part that's way out there is, is adding to the infinite surface area.
Most of the area is out on the tail because if I chop it off, then what, what remains here is, is obviously just a finite area. So it's totally cheating to try to paint the surface by filling the volume with paint because the paint will in effect be spread so thinly.