Transcript:

Okay, here's a question: Can you think of a three-dimensional shape which you can fill? So, if you pour water in it, it will be a finite amount of water. But if you start painting it, it will be an infinite surface. You cannot paint it. Can you think of this? 

The answer to this question is, consider the function 1/x. This is the function 1/x between 1 and, let's say, 50. And we rotate this line around the x-axis. So it becomes like this: It's a rotated shape. Now, this is the open surface; this is the open face. And then the conical surface becomes the solid shape. 

Now, consider what the volume is. The volume is simply the integral from 1 to a, which is right now 50. And then, as this a number tends to infinity, you can see that the volume is simply a finite number of π. 

But what about the surface area? The surface area is given by this. And then, this area is obviously greater than this integral, which resolves to 2πln(a). And as a tends to infinity, the area blows up. 

So, this solid shape can be filled because the volume is finite but can never be painted.