 So, this is the end of our third lecture that's focused on volume, how we can use integration to solve volume problems in a way comparable to using integration to solve area problems. And so, this last thing I want to do before we end this discussion on volume is to introduce something to refer to as the Theorem of Papis. Now, we're going to call this the volume version of the Theorem of Papis because in future lectures, we'll see other things that kind of resemble this Theorem of Papis. So, Papis of Alexandria was focused on these volume problems, so this is the true Theorem of Papis here. And so, let me give you some explanation of what's going on here. So, we have some region that lives inside the plane and it could be a nice, pretty region. It could be some blobby, blobblobblob, you know, gelatinous cube, whatever we want here. But this is our region. It has some fixed area, even if that might be a difficult thing to compute. But it has an area nonetheless. And so, we have this region R. It's in the plane and it lies entirely on one side of some line L. Now, this line L is going to act as an axis of revolution that's going to show up in just a moment. So, we have this line L. What we mean is, L does not cut or transverse the region whatsoever. It could, it could like touch it, it could be tangent to it, something like that would be okay. But no, we'll just assume that the line doesn't cut the region into two pieces. So now, we rotate the region R around this axis L. This forms a solid of revolution. And what, what the Theopapus tells us is that the volume of this solid of revolution, this is going to equal the area, the area of R, area of R, multiplied by the distance that the centroid of R travels. So, what do you mean by centroid? We'll define more specifically what we mean by centroid in the future. But it's like this center, the center of gravity of the region. Now, as this shape gets rotated around the line L, the centroid itself travels some distance. Let me try that again. It travels some distance around, right? It goes on this circular path, something like this. And this circular path has a circumference. And so, how far did that centroid travel as you rotate it around? The Theopapus tells you if you take the area of the region and you multiply it by the circumference of this centroid as it's spun around, that is the same thing. That gives you the volume of the solid of revolution. We'll give you, I'll give an argument on why this thing works in a future lecture. We'll talk more about centroids in chapter eight of Stuart's textbook. So we'll wait and tell then. But this is actually a pretty impressive result right here. I want to show you a quick example. So let's consider a torus, right? A torus is formed by rotating a circle like you see right here. You rotate a circle. Let's say the circle has a radius little r about a line in the plane. And let's say the distance from the center of the circle to the axis is capital R. And so a torus is the solid of revolution formed by spinning a circle around the line. And so it turns out this makes a donut-like shape. So you can kind of see, you get something that looks like this. This circle gets spin around and makes this donut-like shape. And in mathematics, this is referred to as a torus. What's the volume of torus? Well, one way of approaching this is you could treat it like using the shell method or the washer method or whatever. You could look at this right here and you could form an equation y equals the square root of little r squared minus capital R minus x squared. So that'd be the function. And you could try using some cross-sections, things like that going from there. It's doable, a little complicated, but certainly is doable, right? The theorem of Papis offers a very simple alternative to this. Let's start off with the area of the circle. The area of the circle, a, would equal pi r squared. That's a little r, because little r gives you the radius of the circle. And then if you look at the center of the circle, how far does that travel in this journey, right? Well, it travels along the circular path, the radius of which is capital R. So the circumference is going to equal pi capital R squared, like so. I'm sorry, that would be the area of that circle. We would want 2 pi capital R, the circumference of that green circle drawn right there. And so the theorem of Papis tells us that the area times this circumference is the volume so we get pi r little, little r squared times 2 pi capital R. And so the volume of a donut equals 2 pi squared, little r squared capital R. And that's all there is to it. And so if one tried to do this using this technique of integration, using the washer, shell method, something like we were hinting towards a moment ago, that turns out to be a very difficult integral, especially at this point in the semester. Later on we'll learn more about calculating integrals of that form, but that's a really challenging integral. But using this theorem of Papis, it turned out we get a very simple calculation here. Because we are able to identify the area of the circle without calculus, and we can find the middle of the circle, that's centroid without calculus. Now it turns out finding area is typically a calculus problem. Finding the area of some blobby blob, like we saw here with r, that might require a complicated integral. And it turns out finding the centroid of a complicated region also involves using integration. But what the theorem of Papis allows us to do is if we can find area without integration, and we can find the center of mass without integration, then it means we don't need integration to find the volume of the solid revolution. And this little shortcut can be used in other situations that we'll see in the future. If we can avoid using integrals to find area or centers, we might be able to avoid other applications here. And so this technique was first developed by Papis of Alexandria, which was a Greek mathematician who lived in ancient Greece. Well, sorry, the time of ancient Greece, actually Alexandria is in Egypt, right? But Papis of Alexandria was hundreds of years before surizing Newton, right? Yet Papis, like many others of his time, and after before, were doing calculus problems. Calculus is not owned by surizing Newton or Godric Leibniz or some other people we like to give credit to. It turns out people were doing proto-calculus type problems long before the invention of the modern-day integral, right? These volume problems, for example, can we find the volume of a solid by slicing it? This technique was known to the ancient Greeks. And even though they did it with very primitive notions, they didn't use anti-derivatives because that notion wasn't exactly known to them yet. This idea of slicing and having these areas of circles and squares, they could do that and they could stitch it together. And so I do want to kind of impress upon you this historical background that while the fundamental theorem of calculus is awesome, we absolutely can use it all the time, there are many ways of avoiding it. It's not the only way to solve a calculus problem. And this theorem of Papas gives us an alternative that is sometimes available to us. And so that brings us to the end of lecture six. Thanks everyone for watching it. If you like these videos, feel free to subscribe so you can see more of them in the future. Click the like button. If you have any questions, please put them in the comments below. I'll be happy to answer them if you have those questions and we can continue this discussion of calculus sometime in the future. Hope to see you then. Bye.