 Sorry. So if your clipper is eating your batteries very quickly, the batteries are supposed to last the whole semester. So if they're not, you should go to the same site in the library and fix your firmware or don't use your clipper. So apparently it's 26. Okay. Oh. Yeah. So my office hours are in my office typically, which is math 5, 112. That's room 112 in the fifth floor of the math building, which is the closest corner to over here. And actually as we go outside, you can see it, right? So face is this way. On Thursday mornings, which is tomorrow, I have office hours in P143, that's the ground level of the site coming in. That's the plaza level. It's the ground level where physics is. If you come in from the ESS site, you have to go up the flight stairs. It's across from the elevator at 10, 10 to the light of 11, 20. Okay. So if people mostly answered this, yes? Anyone need more time? Yeah. A little bit more? Okay. I'll remind you, we have this fun thing. Tomorrow night, 8.30 PM. And it's in those three different rooms. So most of you are in Earth and Space Level 1. A couple of you are, some of you are in Old Engineering 140 and you're in the other lecture, you're in Old Engineering. Please bring, so as far as the exam goes, please bring your ID, leave a photo ID, preferably your ID card, but at least something with your picture and name on it. Also, while you don't have to do your answers in pen, if you want to say this was graded wrong, it has to be in pen. So I would recommend that, you know, you work out your problem on the back of the sheet and then you write it nicely in pen on the other side of the sheet because, you know, we've had issues where people will say, you know, you're marked wrong and they'll take your paper home, they'll fix the answer and then they'll come back and say, no, no, this was graded wrong. So you have to do it in pen if you want to be able to test your grader. Yeah. Everything we've done is set up as well, you know, stuff. So there is a page on the class website that has five sample exams which will have five different, some of them are exams, some of them are samples, with solutions. Any other, any more questions? Sure. So any rules of all, of various sites, various types, improper in the rules, area between curves, what else did I guess? That's what we've done. And I'm not going to use the graphic number here. No. You have to use this thing. Any other questions about the, yeah. There are nine. You have an hour and a half, there's nine questions. It's like seven or eight in the rules and, oh yeah, Simpsons rule or Trappers rule and stuff like that. So something like that. It's mostly in the rules. So one thing, if you haven't used paying attention, every problem is worth the same amount, whether it's easier or harder. So the first thing that you want to do is look at the exam and decide what's easy. Do the easy problems first because those are easy points. What you don't want to do is find the hardest problem on the exam, work on that the whole time, only get halfway through and then get 10 points out of the 100 and 8, that would be bad. Okay. Other questions? Okay. So it should be done with this by now, I hope. Yeah. Anybody need more time? Well, it's a very interesting distribution on the answers here. I'm stopping both of these. So it seems that people didn't really understand what we did last time. I don't know, 32, 30, 24, 14. So 32 percent, I think this is the answer, 30 percent. I think this is the answer, 24 percent. I think this is the answer and 13 percent. I think this is the answer. So that means a lot of you got it wrong. So what is the idea here? The idea here is that the volume is the integral of the cross section, the area of the cross section. D, the way you're slicing it. In this one, the base looks like, so we just draw the picture again. We have a thing, looks sort of like a wedge of cheese. And when I cut it at a given height, like here, I get a triangle. And it's an isosceles triangle, which means that this distance is the same as this distance. So my picture is, I guess, a little bit, I get a triangle that looks, well, it doesn't have to be that way. I get a triangle like that. And this curve here is y is 100 minus x squared. That means that where is this? This is when x is 0, so this is 100, this is 0, 100, so this height is 100, and this height, or this distance here, is 10. Because when x is 10, we get y equals 0, right? So what is the cross section of the slice? Well, if I cut it this way, it's the area of this triangle. What do I need to know? This is the triangle that's half a square. It's a 45 degree triangle. So right triangle here. What's its area? He knows the area of a half a square, half base type, if the base equals the height. So if we call this distance, I don't know, x. This distance is also x, so it's 1 half, well, x is a bad choice. Well, x squared is the area. That's a horizontal slice. If you slice it vertically, the problem is a lot harder. If you slice it horizontally, the cross section is nice. It's a triangle. It looks like that. And so its area is a half of x squared because this is half of a square. So what is x at this given height? So if I pick a y value here, how do I figure out x is? If I tell you y, how do I figure out what x is? x is 100 minus y, well, that's x squared. So x is the square root of 100 minus y. If I tell you that the height is 7, then the width, the side of the triangle is the square root of 93. If I tell you that the height is 64, then the width of the triangle, what's 100 minus 64? 36 is 6 and so on. So the area is 1 half x squared. So that means that what we want to integrate is 1 half of this square. So the volume is 1 half the integral of square root of 100 minus y squared dy from the bottom, 0 to the top from 100. So that's half of the integral from 0 to 100 of 100 minus y. So that's, now I know some of you did volumes before in high school or wherever, but when you did volumes before, it's quite possible that your teacher emphasized that remembering the formula for volumes of rotation or things like that. I really don't want you to remember formula. I want you to remember why formula are what they are. Knowing the formula for a volume of a surface of a revolution is mostly stupid. To memorize the formula, it's not like two years from now when you're engineering class, they're going to say, okay, this is a volume of surface of revolution, so what's its volume? That's not what happens. What happens is you have something that is you need to calculate a volume or something that's like a volume and you use the idea here, the integral of the cross-section is something you know. You integrate the area to get a volume. That's what's important. It's easy to figure out, if you think a little bit, it's easy to figure out what is the volume of a surface of revolution because you just figure out what does the slice look like, slice looks like a circle. What is the radius of the circle? Okay, we're good. Yay, happy time. Okay? So, well, whatever. So the reason volumes are, even in this class, is not really because volumes are important. It's because the idea that when you integrate, cross-section, you get a volume. That's what's important and surfaces of revolution have easy kinds of angles to figure out. So let me do a few more. For any questions on this, are you okay with this? I know a lot of you are saying, well, this isn't on the exam, I don't need to think about it. It will be on the next exam, so this is about to be so good. Okay, so let's do this a little bit. So at the end of the last class, I mean, it may seem like I'm choosing very easy functions. I am choosing easy functions because, does anyone need me to do that integral? That's like a trivial integral, right? It's 100y minus 1 half y squared, so it's 100 squared minus half of 100 squared, so what's that? I don't know, 100 squared over 2. Well, it's half that, 100 squared over 4, so 25 something. Okay, so let's, so at the end of the last class, I have something like this, which I rotated around the axis to get a horn shape, and I calculated the volume inside the horn. I want to change this problem just a little bit, so I don't want to do this same one again. Instead of this being this horn shape, I'm going to use the same shape, except instead of filling it in, I erased it with, I have this curve y equals x squared. Let's go out to, well, it doesn't matter what we go out to, and another curve like y equals x, and now we want to rotate this area in here around. Again, I will get a horn shape, but now the horn, it's a little hard to see. I get a thing that looks like a cone, well, it's a cone with an inside like that. I think the outside is a cone, and the inside is that volume that we found last time. People understand what this shape is? Okay, so how do we figure out what is the volume of the material here? What is the volume of this area? So, I mean this region, so that is I want to rotate the region between y equals x squared and y equals x around the x axis, and what's the volume? So what do I do? What do I need to know? The point theater segment, okay? Let me just draw the same picture again, and just keep drawing the same picture. Now, what's this point? Zero, zero, what's this point? Where does x squared equal x? One, and what's the y value? One, so that's easy. This is the point one, one, okay? I need to know that. Then what do I need to know? The shape of the cross section. What is the shape of the cross section? Good. So what is the shape of the cross section? Circles. So which way am I going to cut it? This way or this way? Am I going to cut it horizontally? Or vertically? Horizontally? So, he likes me to cut it this way. They don't want me to cut it this way. We can cut it that way. We can do it both ways. Let's do it both ways. Yeah, I'm going to do it both ways, but let's do it. So we can cut it that way, or we can cut it, no, that's the same way. We can cut it this way. Since there's three of them over there, I'm sorry, they're a little cute with a new one. I'm going to go with them. I guess that I shouldn't say. Actually, he's really cute. Are you going to be late? No, sorry. Okay, so if we cut it this way, and we rotate this thing around, we get something that looks like a ring. So let's do this side. First, when I rotate that around, the cross section is a ring. And actually, it's a very, it has a width. This width is dx. All right, so I took this wheel shape, and I turned it, and I laid it down. So I have a little washer. The thickness is dx. I need to know two things. I need to know the outer radius. Let's call it big R. And I need to know the inner radius, little R. So I'm cutting this at a specific x. Let's just call it x. This is x. So when I cut it at x, what's the outer radius? Okay, that's good. And when I cut it at x, what's the inner radius? X squared. X squared. Oops, not big R. So that means that this volume, when I do it this way, is going to be, well, the area of the cross section. So the area is the area of the outer circle, which is pi big R squared, minus the area of the inner circle. All right, so the area of the face of the wheel is pi big R squared minus pi little R squared, which in our case is pi big R is x squared minus x squared squared. So that's our cross sectional area. And we integrate this as x goes from 0 to 1. This is an easy integral to do. The integral is pi times 1 third x cubed minus 1 fifth x to the fifth, which is pi times 1 third. What did you want? Yeah, 1 third minus 1 fifth. So that's easy. So that was not a hard problem. That's not for me. Now we'll go your way, we'll cut it the other way. So if we cut it this way, it's a little harder conceptually, but it's still doable. So I cut it this way. Now what does my cross section look like? No, it's still revolving this way. But for some reason, I wanted to cut it horizontally. You wanted to cut it horizontally, if I could. Right. So what I'm going to get is something that looks like that. It's a very thin cylinder, right? So this is what I get. Something that looks like, that's a terrible thing. A very thin cylinder. Yeah. I don't know. Hey, very thin. You can turn it a little more. Sorry. That's what I get. Now I need to figure out what is, so this is really, when I said it's not really the integral of the cross section, it's the volume of the little slice. This little slice, this times its thickness. In this case, now let's label the bits. This distance is what? Well, when does this curve? Well, first off, notice that my slices go from very thin to very big. I go from a little cylinder to a very big cylinder. The radius of the cylinder increases. Right, so everyone see that? So what do we need to know to figure out the volume of this thing? We need to know it's radius. We need to know it's thickness. So the thickness is easy. That's d y. And what else do we need to know? If I want to find, well, so what else do I need to know? Huh? The shape of the cross section is this. It's not a flat thing. It's a cylinder. I want, I have very thin cylinders, d y thick. I want to figure out its volume. What do I need to know? The radius? Okay, what else do I need to know? If I'm going to make a cylinder out of gold, so it's going to be expensive. I'm going to make it very thin. And I make it one inch high. The radius is, let's say, one. I make it one inch high. Is it going to cost me the same as if I make it 10 feet high? No. So what do I need to know? A height. So what is this distance? Well, so this curve, this curve here is y equals x squared, which is the same as x equals y, as you said. And this curve is y equals x. So this distance, as you said, is square root of y minus x. I'm square root of y minus y. If I cut it at a height of y, I'm going to get a cylinder whose height is square root of y minus y. So this distance is, well, this point here is the square root of y, and this point is y, so the height, well, it's really the width, but square root of y minus y. And what's the radius? Does nobody have a clue what I'm doing? Okay, this one's easy. The other one's okay for this. Okay. Let me draw this picture again. Look, you have to know how to do this. It'll probably be on the next exam. Won't be on this one? And you will definitely have homework problems. You just did the easier one. Okay, you want me to give you the only way it works, is the hard way? Okay, so I'll do that next. There are some where the hard way is much easier than the easy one. So the hard way isn't hard once you understand it. So let me just finish this problem. Okay, it's just that it's not clear to me that you understand what I'm doing. So we have a little cylinder. I want to figure out this cylinder. So I have a cylinder. Let me turn it sideways. Looks like this. You know what we're doing now. It's a cylindrical shell. Yeah, now you know what we're doing. I have a cylinder that looks like this. It's height is square root of y minus y. It's radius here is y. It's thickness is dy. And I want to figure out the volume of this little slab. Well, the easiest way to figure out this volume is to actually cut this thing open, lay it down. So I have a cylinder here. And I cut it open. And I lay it down. And I get a shape that I know of that. I have a heart, a difficult object whose volume I want to calculate. It's thickness is dy. So it looks like this. And I cut it. And now I want to open it up and figure out what it is. So when I cut this thing open, I get a little piece of paper. It's a rectangular slab. Yeah, that's what I'm trying to say what that is. When I cut this open, I need to know this distance around. Well, this distance around is the length of this. Well, I cut it open. So this distance is 2 pi r. r is y. So it's y. And this distance, since I took this guide and laid it this way so I can look at it easier, this distance is square root of y minus y. And this thickness is dy. Now, yeah. I just want to understand why square root of y is y. Okay. Well, I know you're going to say no, I'm going to draw the picture again. But I'm going to draw the picture again. We have a thing that looks like that. Where are you? There you go. Okay. And the thing that looks like that. We're going to cut it. This way. And I get a little thin piece. I revolve that thin piece around the x-axis. It gives me a cylindrical shape, which I'm going to pick up a little bit here so I can look at it vertically. Because it's easier to draw a cylinder vertically for some. Okay. This piece is at height y above the axis. So the radius of the resulting cylinder will be y. If I tell you y is a half, you know the cylinder. If I tell you y is a quarter, you know the cylinder. I guess when I revolve, it will be bigger than the one I've written. So when I revolve this around, I get a ring that goes like that. How tall is the ring? Well, it's the distance between this curve and this curve. This is the curve y equals x squared. And this is the curve y equals x. So if I tell you y, it's one quarter, and I want to find the distance between here and here. Well, this is the point, one quarter, one sixteenth. And this is the point, wait, sorry, y is one quarter. So this is the point, one quarter, one half. And this is the point, one quarter, one quarter. I tell you y to know x, you must take the square root. This is the same as x equals the square root of y. I tell you y is a half, that means x, I mean, I tell you y is a quarter, that means x goes a half. So I take the square root. For this curve, I tell you y, I've told you x. The distance from y goes a quarter. The distance from a half to a quarter is a half minus a quarter. The distance from this curve, which is x is square root of y, this curve, which is x is y, square root of y, y is y. Does that make sense now, sort of? You know, just rather than that. So that's the size of the thing we want to integrate. Yes. Think about the shape. The shape is the cylinder. I don't want the volume of the cylinder, I want the area of the cylinder. The surface area of the cylinder, 2 pi r, r is y, because I'm going around the x-axis. 2 pi r, the height is the distance between curves. Yeah, but that wasn't what the question was. No, I'm going around the x-axis. I'm doing the same problem I just did, a hard way. Because we wanted to do it the hard way. Because we've got to do it the hard way. That shows you, like, much. Okay? So this resulting integral, so I know this is a little confusing. Resulting integral, I integrate from 0 to 1. I'm letting y change. So I'm going to integrate p y. This thing's going around here, so I'm doing horizontal slices. And what I integrate, 2 pi r, the height here is root y minus y. And I'm integrating it d y. So this is the integral 2 pi. So y root y, which is y to the 3 halves, minus y squared d y. So this is 2 pi times 2 thirds of y to... No, I'm sorry, what am I doing? Y to the 5 halves. So it's 2 fifths. Y minus 1 third y cubed from 0 to 1. Uh-oh. What did I do wrong? So something's wrong here. Yeah, it is. What does it say? 2 fifths minus 1 third times 2. It's the same as 1 third minus 1 fifth. I think I screwed up. So this is 5 fifteenths minus 3 fifteenths is 2 fifteenths. And this one, 2 fifths is 10. No. 6 fifteenths minus 1 fifth. Hold on a second. Good. So this is still 2 fifteenths, 5. So there's a second. OK. So this is a... Yeah. With the bottom... No. So let me change this problem a little bit. Can we do a different problem? Can you say that it's a problem? So he has a urgent question. So yeah. Because the cylinder changes as I move up and down. This is the volume of the... This is the surface area of the cylinder. The volume... It's not the volume of a filled cylinder. It's the volume of the thin cylinder. All right. So I'm going to make a sculpture. I'm going to make... I'm going to make a piece of pipe. I'm going to make this piece of pipe out of lead. And it's going to have a diameter of 2 feet. 6 feet. How much lead do I need? And a width... I'm sorry. A width of 1 inch. How much lead do I need? That's the question we're asking. I'm going to need one-twelfth of diameter of 2 feet. So the radius is one foot. So one-twelfth of 2 pi out of 6. 2 feet. That's what we're doing here. This is the height of the cylinder. This is the circumference of the cylinder. This is the thickness of the cylinder. What is the volume of that object? It's a thin thing. So her question. You always go from the thing you're revolving around. It depends on what the question is. Suppose that in this case, instead of making this morning shape, I'm going to cut the end off. So the thing that I'm going to revolve, the thing that I'm going to revolve, might look like... It might revolve a shape like this. Well, here I'm going to revolve around this line. So the inner radius will not go to zero. So I can't start at zero. So I don't go all the way down to zero. Okay. Why did I do this? I did this because I wasn't planning to do this, but he gave up and asked me. I don't know. Because he said go the other way. You could do this either way. If you are not a masochist, you will do it this way. But if you're feeling masochistic, you might do it this way. However, there are some shapes. For example, suppose I had something that looked like this, a bump. And now I'm going to revolve this bump around the x-axis. And let's say this bump is not symmetric, so the bump that I'm using is 2x squared minus x cubed. It actually looks like this. So say I have a bump like this that I'm going to revolve around the x-axis to get something that looks like the y-axis. Sorry. Whatever his name is today. So I get something that looks like the top half of one of those donut beaches. Right? It looks like... Does anybody know what I'm talking about? The donut beaches? They look like, they look sort of like the top half of the donut, but the business... Now that looks obscene. Okay. I get something that looks like a bowl. But the bowl is a big, right? If I look sort of from the top, it has a dimple in the middle and it goes around like that. Like a butt cake. Like a butt cake, but the butt cake doesn't have the hole. Right? This comes in and fills in the hole with the butt cake. So I made a butt cake, but I overfilled the mold and so like I made some little cake in the middle, too. So it looks like that. Sort of like a butt cake. Now, if I try and slice this thing this way, these slices are nasty. Because even though the slices are nice rings, so if I try and slice my butt cake horizontally, my cross-section will be a ring that's easy to figure out. If only I know where this is. But solving this equation for X is nasty. It's doable, but it's nasty. I don't want to solve this equation for X. Well, it's much harder to figure out where these points are. So this problem is much easier if I use a hole saw to cut up my butt cake. I take a sequence of little cylinders at various heights to cut up my butt cake. I have a stack of rings to cut it up. So in this case I take my cross-section like this and I revolve it around the wire. So in this case it's actually much easier to cut it vertically to do your volume by shelves. That's the secret word. So this one's hard. This is the cross-section. But the radii are tough to find. So I need to solve Y equals 2X squared minus X cubed for X. It's icky. It's a formula, but it's nasty. I don't want to do it. So instead I do it the other way. And that means I do it like this. So this problem is actually if I slice it let me get rid of that. If I slice it the other way if I slice this thing the other way if I pick an X value and revolve this thing around the Y axis it will give me a cylinder that looks like that. And this is a piece of pipe. I want to know how much stuff I need to make this piece of pipe. So the radius is in this case X. It's the distance from here to here. My height is just the value over X. It's f of X. f of X is 2X squared minus X. So the volume is well, my X is going to start here and walk to here. So my volume is the integral of Y times the radius. That's the circumference of the resulting cylinder times the height of the cylinder which is 2X squared minus X cubed and the thickness of the cylinder is DX. This is not a hard integral. The other one is nasty. This one's not hard. And X starts here at zero and it goes out here and this thing hits the axis so I need to know when is 2X squared minus X cubed equals zero. I can factor out an X squared 2 minus X so it hits it two. So I integrate from zero to two. That's actually much easier than trying to make a washer. Because finding the radius of the washer is tough. Now I was going to use this one to introduce this if you wanted to do it the hard way. So we did it that one both ways just when I'm not going to do the other way. So the general theme here is the same one to find a volume. You slice it up into thin little things and you calculate the area of the thin little things times their thickness. Here my thin little things are pieces of pipe and I have to think a little bit to figure out what their area is. But it's not hard. Okay, that's it. See you Thursday night on my own TV4.