 So what we've done so far is we've talked about how to do integrals, what they represent, and then the next three or two is on how to use integrals to do various things. So one of the applications that we're doing is finding volumes. Now you need a little more generality than we have for this class. And it's actually in that two-row calculus three that give you the general version of volume. So we can do volumes of some very regular things using the tools we have already. So for example, things you already know, without immigration, if I have a rectangular thing that is with height, no, height is usually up. With length and height, what's the volume of this? Length and width and height. Suppose I make it a little different looking. Suppose it's a cylinder, but this ring is R, and this time is still H. What's the volume of that? So what is the same? What is in common with these two things? They both have a height. Okay, that's good. There's a relationship between this pi R squared H, and if I take the saw, cut the top off. Or if I take the saw, cut the top off, look at the area of that. This area is length times width, and this area is pi R squared. And it's no accident that the volume of this object is just length times width times however tall it is. And the volume of this object is h times the area of the cross-section. In fact, that's generally true. If you can calculate the area of the cross-section of something, then you know it's volume. If you know the area of all the little slices, then you know the volume. And of course, I wrote this on one of my temperates. I'm going to leave that. I guess I'll move over that here. So suppose we have something slightly more complicated. Suppose we have pyramid like this. Let's make it square. So it's 3 by 3, and its height is about 5. How would we figure out the volume of this? Yeah. This has 4. I'm not asking about the surface area. I'm asking about the volume. How could we figure it out? So the area of the base is 9, and the height is 5. Do you think it's 45? No, because if it were 45, that would be the area of anybody who would look like this. And that would be the volume of something like this, where this is 5, and this is 3. This would have volume 45. So, trying to generalize this notion, don't tell me what the answer is. Tell me how we might figure it out. This idea. Area of the base is 3 times height. Why would it be that? Why would it be that? As you go further, do you think it's the area of the base? Okay, so let me make you write. Suppose instead of looking like that, it had, is it terrible? Let's try it again. I want to make it sort of have a parabolic slice. So now I want to make it look like a parabola in the bit. I want to make it pointy. You would establish that this would hurt. Again, this is 3 times 3. So I claim that the same idea here works here. What is the idea here that works here? Yeah. Well, it's not times the height. Yeah. So the idea to think of here is we take. An integral is a way of adding up lots of little things. So if we integrate area of the cross-section, the height, from the height goes from 0 to the top. I guess these are both 5 times. This should be the body. So in this case, that's exactly what we did. The area of the cross-section is always pi r squared. The height goes from 0 to, I don't know, 2. So this volume is the end growth from 0 to 2 of pi r squared dh, which is h pi r squared from 0 to, I don't know why, 2. Whatever. Which is h pi r squared minus 0. This gives us what we had before. If we integrate this thing from the bottom to the top. Here, the same. I'm going to integrate length times width dh, and that gives me h times length times width. It's the same. I'm adding up these little slices. Here, it's a little more complicated. The slices are not always the same size. If I slice this thing down low, so here, if I slice it too big, up here, my slice is small. And here, my slice is a point. The idea is the same. The size of the slice depends on how high up I cut it. So the size of the slice here is some function of the height. And if I figure out what that, and the same thing is true here. Here, my slice is big. Here, my slice is small. Here, my slice is tiny. Again, the size of the slice depends on where I cut it. Unlike these guys, where the size of the slice does not depend on where I cut it. So now, maybe we could look at this a little more like the way that we set up an integral. We have some object here that looks like. I have some object here. And I want to calculate its volume. So I think of what I, in order to calculate its volume, I cut it up into a bunch of chunks. It's called a snake. It's a big, really big thing. It was a candle, and it was hot, and it's really melted. So I cut it up into a bunch of little cylinders, in this case. And I add up all of their little volumes. So, I don't know, sometimes you may have seen where you make things. But actually, how many of you have been to or heard about the rapid prototyping lab they have over an engineer? It's a 3D printer. People know about 3D printers? So what a 3D printer does, so it's a big machine or a little machine if you buy a cheap one, then what it does is what it wants to make an object. So there's also something called a maker box. In order to make your box. No, it's a 3D printer. It's an object that you give it instructions to make a thing that you can hold in your hand. And what it does is it lays down a layer of stuff. It prints a little layer of stuff, because it squirts out plastic or nylon or the fancy one that they have over there has a power that they stack with the laser and the power melts. And then on top of it, it makes another layer of stuff. And it builds this thing up layer by layer. And in the end, you have an object that you can hold. You have to go pay to make it. So if you're an engineer and you take this class, then you can make stuff. Like right now, go over there and say, make me a thing, then they will charge you a fair amount of money to make you a thing. But if you have the thing, you can go over there and what's the guy's name. It's over in heavy engineering. You can say, hey, I heard about this graphic prototype. It's way cool. Tell me. Maybe they'll show you. No, it's cool. But yeah, it's expensive, because the big one that they have that zaps the power with lasers, it takes about 10 hours to print, you know, an old print run of volume about this big. And things are expensive. So they would charge you a lot of money to make stuff. About a lot. I don't know. But anyway, it does it by laying down layers of stuff. Well, we're doing this backwards. We're computing the volume by slicing this up and then computing volumes of little things that add up to the big thing. Well, this is exactly what we did when we computed areas by integrals. When we computed areas by integrals, we have some region. We slice it up, compute the volumes of the little rectangles and add them to the integral of f of x. The x is the limit, zero is the sum. The same thing here. The volume here, as the height, the little ph goes to zero. So the volume, the height, the delta height goes to zero of the area of the cross-section times the height. Well, this is an integral. So this is exactly the same thing that we've been doing all along. So now let's do one of these. So here I have this pyramid and I want to figure out the area of the cross-section as a function of the height. So what do I need to know to know the area of the cross-section? I know the area of the base is nine. Suppose I want to know the area at height three or four. What's the area at the height four of this pyramid? I mean of the cross-section. So I'm going to take this pyramid, I'm going to cut the top off four inches from the bottom and I'm going to look. I have a thing. When I cut the top off four inches from the bottom that looks like this, what's the area of that? How can I figure it out? Tell me a number. You don't know. I can figure out this number. How do I figure it out? Whoa! Zero to five of... Well, I have to know what to integrate. Maybe you don't understand my question. I have to know what to integrate. So what does the area of this cross-section depend on? Okay, forget about heights for a minute. Suppose I cut this thing and I look at it. It's a square. What do I need to know? Well, it's a square. The side. I need to know the side length. Okay, the side length depends on how high up I am. Right? The side length at the bottom is three and the side length at the top is zero. What's the side length halfway up? Probably one and a half. What's the side length four-fifths of the way up? It's three. How can we be sure? So I don't know... So the side length depends on the height. If I look at the bottom, I know that s of zero is three. At the top, I know that s of five is zero. What does function s look like? So this thing is a straight line. This is a linear function. So we need a line. This function s, putting it at the height when I get the side length, when I put it in zero, the answer is three. When I put it in... Oops, sorry. At height zero, s is three. At height five, s is zero. This is a straight line because the sides are straight. So what is the equation of the line? It goes through zero, three, and five, zero. It is... So here s minus... So the slope is negative three-fifths, five-thirds, right? Five-thirds. So it's five, negative five-thirds. Three-fifths. Yeah, three-fifths. One of those. Yeah, it's like... Yeah, it's like... So when h is five, you get s equals zero. When h is zero, you get s equals three. Okay. So that's our function. Right? That's the equation of that. One is what? An integral from something dh. What's the lower line? Are people following this? I'm seeing a lot of blank lists. Anybody have a question? You're just writing here. It goes from zero, right? The height starts at zero. What does it go to? Five. And what do I integrate? This? That's the integral I want? The whole thing squared, because the area is squared. So that's the volume, which I guess you could push it around. So this is nine-twenty-fifths, integral zero to five, h minus five squared, which is twenty-fifths. Why can't I do this? h minus five, q over three, which is the value I get from zero to five. So that's a hundred and twenty-five, a hundred and twenty-five over times nine times three, which is what? Why can't I do this? Nine over three is three, five, fifteen, which is what the guy said before, because he made the volume of pyramid with one-third of the volume of the frame. That's one-third of forty-five, so it's right. Okay, so this one was relatively easy, even though you'll see that you can read by it, but okay. So now let's move on to the next one, which is the same thing. So I want to do the one that I erase. So now instead of having my pyramid be a straight-sided pyramid, I want the sides of the pyramid to look like this, like the cross-section, the vertical cross-section to look like this, where this is the parabola, let's do, so let's do it. So instead of three, let's put this to four. So this is, why can't I do this? X minus four, that's square. Is that work? No, four minus X squared. That's this one. This one is, so this is my cross-section, and the base, so at the bottom it's eight by eight. How should I make it? Eight? It's whenever this gets zero. So how do I do this one? So the height function is zero, it's eight by eight. Sorry? When X is zero, how tall is it? Sixteen. Sixteen, right. When X is zero, we get sixteen. So this height is sixteen. Is it clear the thing I'm computing the volume of? It was this guy that I turned to here. Before, except it's gone three by three, saying by eight now. So what do I do? What I need to figure out is the side length at a given height. If I tell you H, what is the side length at a given height? So the thing that I need to know is if I tell you at a certain height, and how wide is this cut, how can I figure that out? If I write X in terms of H, which is an idea. If I write X in terms of H, so I have, what can I do? Square root of H equals four minus X. So X four minus square root of H. That's this side. And here our height is positive, so that's fine. And then this side, I can do the same thing. I get square root of H is four plus X. So X is four minus square root of H, standing in front of me. Oh, negative X is that's twice that. So this guy's X is that one, negative. So the width H is twice four minus square root of H. So if I tell you that the height is 10 that's four minus square root of H. 10 then the width of this thing is two times four minus square root of H. If I tell you that something is wrong here. Oh yeah, the height is 16. And the width is two times four minus square root of 16. Four minus square root of 16 is zero. So two times zero is zero, so that's good. Okay, so now I know the width. So what's the area of the cross-section? No, I don't want to integrate that. So what does the cross-section look like? Yeah? Yes. The cross-section in this case is a square. It's width is two times four minus square root of H. And it's height is two times four minus square root of H. So it's area isn't a square. We can figure out because two square is four. I'm not sure which H you have here. We can do that. It's the whole thing's square. So this is the integral from zero to 16 of four times four minus root of H is square root of H, which is some crap for you. So here square this out. 16 minus 8 root of H plus H dH 16 H minus 8 H to the 3 halves times 2-thirds of H squared 2 from zero to 16 which is four. I don't know what 16 squared is. H to the 3 halves that's 4 Q so that's 64 which is some stuff. So that's 3 house, I don't know what it is. So 16 times 64 is 16 squared, right? 4 times 16, I can take that, 4 times 16, and I multiply it by 8, so I can pour one of those 4 into 16 squared, so that's 2 times 16 squared, 4 times 16, 3, this is it's 3 1⁄2 to 16 squared here, minus, and then this is 4 3rd to 16 squared, right? So 3 house minus 4 3rd is, I'm sorry, 6 9 6 is minus 8, so it's 1 6, so it's 4 6, 16 squared, but whatever that is, sorry, it could be written in English, it's not my problem here. So the idea here is, in order to find volumes, you integrate cross sections. So that's the whole point here, is that volume is the integral of the area of the cross section. So now let's move on to something else, which is the same but different. The area of the cross section, whatever it is, is happening to be a square or not a square. The cross section in both of these examples was a square, so I found the side link and square it to get its cross section. If the slice was a triangle, then I would need to find the area of the triangle. If the slice was Mickey Mouse, I would need to find the area of Mickey Mouse, just to picture it. I can't draw Mickey Mouse. It's a cat, I can draw a cat. So if the cross section is a cat, you find the area of the cat. Yeah, the eyes are all over, you cut the eyes out. Every time I see a cat, I can't decide. My cat doesn't like it. Okay, so the general thing is that the area, the volume is the area of the cross section. That's the thing to take away. You figure out the volume, the cross sections are done. Now, often these are done in terms of saying I'm a mouse, instead of having a pyramid thing and it's full has a parabolic side. So this is a parabolic dish like this. You want to find, you have a, you find a parabolic dish that's, you know, from somebody's satellite hookup and you want to fill up the water and don't have much water involved. They probably wouldn't appreciate that. Okay, so we have a parabola here and we rotate it. So let's just do the parabola y to the next square and we rotate it around the y axis and this gives us a bowl. Let's say the parabola just goes from zero to one. What's its volume? What is the cross section? The cross section is a circle and you have a radius and it has a radius. What is the radius? So I have to think about where I'm cutting it. What do I know when I cut it? Do I know the x value? Do I know what the y value is? Well, you can know either one. If I know x here, then I should know y. So what is the radius? So the radius is, so this is y is pi, so r is not times. r is x square, that's fine, but I don't know x necessarily. So let's just say r is x square for a minute. And so then what is the volume? Well, it's not useless. So since I'm rotating around the y axis, I'm going to want to integrate dy. So x square, if I know y, I want to know the radius. Sorry, the radius is not x. I'm just wrong. So I need to know the radius y. That means that I look at this parabola and I tell you the y value. I want to know this distance, right, square root. So since this is y equals x square, that means x is square root y. And so this distance is square root y. I'm going to rotate it around and get a disk radius square root y, which I now integrate. So the area is pi square root y squared, which is just pi times y. So that means that the volume is 1 from 0 to 1. Pi, we can pi there. Pi y, well, do y. If you want to call it h, then you have to pi hdh. So this integral I can do, pi over 2 y squared, we can evaluate it from 0 to 1, which is pi over 2. The volume is pi over 2 y squared. So that's not too large. Suppose we decide to get weird and rotate it the other way. So instead of making this bold shape, I want to make a horn now. So I'm going to take this and rotate it around the x-axis, the horn. So I'm going to rotate x-axis to get something that looks like that. So what do I do now? I put it in terms of x. Now my cross section is a vertical circle. And again, this is still y equals y squared. If I cut it somewhere like here at a given x value, then I need to figure out what's the height. What's the height? x squared. So here at a given x value, cross section has an area, so it now looks like x squared is my radius. So the area is pi x squared to squared. So now I want to integrate for this form, I integrate 1, pi x to the fourth, dx, because my slices move from here to here. And this is from 0 to 1. So that is pi x to the fifth over 5, 0 to 1, which is pi over 5. Yeah. So here the lower and upper, so I made, you know, this parabola goes to 1, 1. Here the lower and upper values are y equals 1. Here it's x equals 1. So if instead of doing, let's just change this problem a little bit. So if instead of going to x equals 1, let's go to x equals 2. In that case, if I go to x equals 2, this volume goes to 4. And I get 8 pi because I have 16 pi over 2. And if I go here to 2, then this is x equals 2. So then I get 2 to the fifth, which is 32 if you want. So next time I'll do it, if I don't, if I prefer to do this in terms of x instead of y. So what I want you to do is to think about what the cross section looks like and figure it out. So I can just tell you the formula, but if you want to understand the formula, you get the formula. Understand the cross section. Okay, see you to Wednesday.