 I'm going to remind you that in these things that we use every time and time, if you don't have one and you have a smart phone or a laptop or something like that, you can use that instead. I showed how many people do not have a computer but do have a web-enabled device or some sort of event. Excuse me. I'm going to show you what I'm going to tell you. If you go to the class page, you should either get to the blackboard or go to this web page or there's a link on the front that says something like for clicking without a clicker or something. Okay? So you can just open that page and then there's a link about it. So in fact, since you have your clickers here, let's have our first clicker question. You can still do it. Okay? So the way the clickers are graded, by the way, is you get one point for answering and one point for answering correctly because if you have, so, you should be able to figure out the difference between these two things where some of you, this is the right answer and for some of you, this is the right answer. Yeah. Oh, yeah, because I'm stupid. Thank you. So for those of you who don't have one, I screwed up. Thank you for reminding me at the launch of web browser and starting the question. For those of you that don't have clickers, it will say there's no question open because the network doesn't like me right at the moment. But it will. So please be patient. Click the try again button because you can give them a choice. Come up. There's the question. It just says there's a clicker. You don't have a clicker and you're using a smartphone or something. You should have a bluish page and it should say answer your answer and also enter your ID number. Okay. Did everybody manage to figure out to answer that? You have 97 responses. You have clickers. You just press the number and then you press one, two, or three and you can press the little thing in the center that says send. And then you have a little smiley face. It has a little smiley face. It means I got your answer. For those of you that went home that should say I took your answer. Okay. Everybody's answered, right? Except him. He hasn't. We'll get a chance later. Okay. So I'm stopping this question now. So I can see here. I'm not going to put it up unless it matters. 99% of you answered number one. 1% of you answered what? I don't get to see your results until I go back to my office. You should have all answered no. If you answered yes. Well, I guess yes could be correct. It just means it's not with you. But anyway. Okay. All right. So I will do this. A lot of stupid questions about this, too. As we go. I won't necessarily ask clickers questions every five minutes. I may only ask 1. I may ask 0. I may ask 10. Whatever. The point of this is to just get you to interact with each other and stuff. Okay. Yeah. Okay. So now we need to do some calculate. Well, there was one other administrative thing. And I just forgot where it was. Okay. So a web assignment homework. There is a homework assignment on a web assignment. It's up. It's been up for a while. It was due on Wednesday at 9 a.m. Because of the hurricane, I added a couple of extra days. It is now due far too early Saturday morning. This means do it before you sleep on Friday night. If you don't go to sleep on Friday night, I'll let you do it on a Saturday morning. Do it. If you do, any questions that you answer, original due date, give you extra credit. Plaint this last time. So don't be confused. The way the grading works is if you answer right the first time, you get full credit. If you answer right the second time, you get one half credit. If you answer right the third time, you get one third credit and so on. But if you answer right the second time but early, you get one half credit because you get a bonus on your half credit. You get three quarters credit. Yeah? Like if you answer right the third time, like if you answer right the third time, you get seven days before. No. If it's more than two days before the due date, you get the credit. You get the bonus. So you don't get a bonus over doing it right now versus waiting until Monday. Generally, if you do it over the weekend, because there's usually one Wednesday, you get the bonus, but if you wait until Monday or Tuesday, you know. So the main point of the bonus is to get you to think about the problems, do those you can do, and then worry about those that you have trouble with later. I know there was another, but I can't remember what it is. So is there anything else? You don't know what I covered. All right. By the way, this is Professor Bonifam. I don't know why she's here, but she is. She teaches the other lecture at 5.20. And for some reason, we don't want to come to this lecture any longer because I see two people are going to, two disorganized go to her. I'm going to do some math. I know it seems confusing, but we'll do it anyway. I don't need to say that. So, but you can tell me that. Not really. So he said study abrasive chain. That's not wrong. Don't feel good when I'm trying to get here, but. So what calculus, anybody else have an idea? So we're writing them off. No? Nobody knows what calculus is. You've all had at least a semester of calculus and you don't know what it is. He knows. He's the only one that's been to class. Yeah. Okay. Or rock. Yeah. So, okay. So what it is about, I mean, the main, the big idea in calculus, the huge idea in calculus is that you can understand topic stuff. This is. That's the big idea. This is the big, and then why didn't it have. Newton was supposedly sailing on a river and he wanted to understand how his boat was floating on the Thames or something. And he wanted to understand why his boat was moving and he was looking at the currents and realized that the currents were pushing him and if he just knew how all the currents fit together, he would know where his boat was going to go. And this is the big idea that makes calculus work. And the way that we understand this microscopic detail in modern days is via the notion of a limit. You know, in many, many high school classes and so on and acting college classes, a lot of students whose idea of a limit is that stupid thing that you have to do at the beginning of the year before you can get to doing the real stuff where you take derivatives by formula. But for mathematicians, this is the thing. This is what calculus is. If there's a limit there, it's probably calculus. Mathematicians tend to call this analysis, but if there's a limit, there's probably some calculus hiding somewhere. Name. So these macroscopic things that we want to understand, so we have some function, some function. We want to know stuff. We want to use this microscopic detail to learn things about it. What you all studied in your first semester, one of the things that you studied is the notion of the derivative. The microscopic thing that we're looking at, we take our function, let me do it over here, and we blow it up. It's pretty straight. It's not quite straight. A lot more. It looks like a straight line. So this is the idea of blowing something up and zooming in until it looks straight. What are we doing when we're zooming in? We're taking a limit. We want to understand what's going on at this point. We understand that. We zoom in, zoom in, zoom in. And this process can be formalized. It goes to zero plus H. Maybe let's call this A. And this gives us a thing that you know about, but you probably forgot this formula, called the derivative. This is not what we're talking about in this class, but this is what we will use all the time. And so this is one of the big ideas of calculus, is the idea of a derivative. What's another big idea in calculus? Okay, so integration. The idea of integration. The idea of integration, we want to ask a slightly different question about the function, not how is it looking in the small. We maybe assume that we know something about what's going on in the small. We want to know for the same function. And we want to use the same idea. And in order to understand the big question, the macroscopic detail, we want to focus on little ideas to put it together to understand it. So in order to find the area here, this is my graph. We want to find the area there. And something that you all should know, and if you don't know, then you should be in math 126 instead of 132, is the way to find this area is little pieces. I'm going to use big pieces, but let's just chop it up. So I chop it up into pieces, and instead of finding the area, it looks like something that I can take the area up. So maybe I chop it up a little more. If I keep on chopping, the pieces are, then they have some little bit of curve on top of them. Curves tilted down or tilted up, or whatever. I just think this looks, I mean, to you, I can see that I do the top tilted. But I doubt any of you can. Maybe you guys in the front can. So this is a rectangle. Just like in this case, this bit that's curved, if I chop it tiny enough, it looks like a rectangle. Rectangles are really easy to find the area of. You just take the height times the width, and we know the area. And so if you chop this thing up, and you do tiny enough pieces, all we have to do is add up the area of rectangles. And that gives us the area of the big thing. Nobody knows really the height times the width. Well, we'll stick the widths for a minute. What's the height? Well, depending on where. But if this rectangle sits right here, point here, let's call it x star, sitting in there, then what's the height? If I put n rectangles here, and I make them all equal, so if I used, let me use a number. If I used a hundred rectangles, yeah, it's not a plus b divided by a hundred, because that would say this distance is over a hundred. Instead, it's this distance, which is this minus that, minus a over n. Now, about, I don't want you to memorize all sorts of formulas and stuff. I want you to think about what's going on. If you try and do this class by memory, you either have to have an amazing, even if you have an amazing memory, next year or the year after, you think about what the idea is, but fit together with the idea, this class is not so hard, easier classes, but that's the way to succeed, is understand what's going on, and then the memory becomes easy, because the memory is, oh, it's kind of like that, oh, it's this because that's how it works. If you just memorize formula, you'll make little mistakes, like a plus b over a hundred, rather than b minus a over a hundred, or you'll remember a wrong formula, and you'll screw up. So, this is the width b. Well, we take a hundred points, scatter them x2, x3, I'm going to forget about the star, up to x100, soon I'm going to change the 100 to an n, is going to be, well, I take the height of x1, half of x3, blah, blah, blah, I'm not going to write all of it. This is my approximate area. It's not the actual area, but it's pretty darn close there's a hundred little whiteậtish there, they're all very squint scheme. It's really close. The key is to write I, most of you, you would say notation, anybody not familiar with some notation? It'd be, you're not familiar. Okay, so, I will now introduce it here. until October. We'll use it a little now and then we'll use it a lot in October. So this is tedious to write. And if instead of 100 I had a thousand it would be different one word of change. This means this is a Greek capital S standing for sum and the things we're going to use again somewhere else. Starting from 1 and here going to 100 and what our mission is exactly what is in the parentheses. So these are exactly the same thing. There's no math here, there's only notation. It's exactly the same way to write that. So this big sigma shouldn't be scary if you program this is a for loop. So this is for I from 1 to 100. What level should I write it? I don't know. Basic. U sum equals f of the people old sum plus f of x sum i. And do. There we go. So this is just take this thing at it. Take the next thing at it. Take the next thing at it. We will study these in much greater detail in four weeks If we change into it, let's put the sum on the outside. Now we haven't used this idea of limit yet at a picture at this level. The area is very close to something straight but it's not exactly equal to something straight. So instead of taking the limit sort of at once the thing the rectangles get small and the number of them gets big. It's going to be just about b minus a over n. You should spend a lot of time on that. It's going to be less than the sum of this form depending on how we choose the next i and bigger than another one and these two things go together. Let me draw that picture actually. So let me do n equals 4. Here's my function. I want to go from there to there. I want the area. I'm going to chop it into four pieces. There we go. I'm going to tell you how to choose these x i's except that they live inside this region somewhere. So I get to choose x one in here somewhere. If I choose it here, this is x one. Then I choose x two. Well it looks to me like I want the lowest. Here this is my x two and then I'm going to choose it here. This is my x three and then I'm going to choose it. It looks like about here. There's my x four. Then the area I will get will be definitely too small. Absolutely too small. It can be no bigger. I mean no smaller because every rectangle or the big rectangles in this picture sit underneath the curve. So this is a low. It is for sure. Even if I know nothing else about the function except the value here, here, here and here, the area is for sure less than what I get with those four rectangles. Bigger. Those four rectangles are less than what I get for the area. Different game, let me just draw the same picture. We're pretty close to the same picture here. So if instead I chose v, I'm going to call them m, m for maximum input. Whatever. I will call them v. If I instead choose this point to be my first one for this rectangle and then I choose the same but this point to be my v two here and then I choose this guy, be my v three and then I choose this guy to be my v four for sure too big. It should be reviewed to all of you. So that means that the area for of v minus a over four goes to two numbers. Function looks like the one that I choose goes according to the rule I chose. The x i's are the minimums and the v i's are the maximums. I could have done the same thing with 40 or just some number n. You'll still get the area is trapped between those two things. But notice also that if I use lots and lots of rectangles and the function isn't too crazy, this will get bigger. If I use twice as many rectangles, if I use twice as many rectangles, then I get a taller one there and I get a taller one here and I get a taller one here and I get a taller one over here. So I would get a bigger area on the bottom and a smaller area on the top. And so as we take more and more area, we'll do the same thing. So if we take the limit by choosing them, then this goes to some number. So if this number exists, this is the area. This is the definition of the integral. If we chop it up into a bunch of rectangles, choose a point inside the rectangle and take the limit as the number of rectangles goes to infinity and the width goes to zero, this understanding that when it comes down to looking at little slices is extremely important for understanding what we do later. So even though using this formula, if you think back to when you did calculus one, you used this formula for a couple of weeks or a month or whatever, and then your professor showed you a magic trick that you never need this formula anymore. And you said, see, why did I sweat so hard to do that formula? And similarly, probably made you, magic happened and there's an easy formula. What I like to do in math is take a hard formula and then make it easy. If we start with the easy, then when we try and apply it to something else where the original problem doesn't fit, it's not going to work. You don't know how to adapt it. So it's important, and we keep emphasizing this definition because this is what's really happening. It's like if you have a car, if you're not an airplane, how about a boat? The same principle that makes the boat motor is the same. If you have no idea what's going on in a motor, the motor boat is the same. There's little explosions going on in the distance, jumping up and down and all that. And we want to change this to make our motor not calculate areas, but calculate something else. Emphasize this because if I just start with, which you can probably all do, what's the integral of x squared, you all know that it's 1 1⁄2 x cubed. One third. So you all know how to do this, or just remember, we got it with something to do and I believe it is. That's why I named it 32 instead of 126. You have to make me do this. It's reminding you, so integrals. We know some integrals are areas. Well, one observation that we can make, this picture, is if the function is positive. If the function is negative, it's not really an area. It goes like this. Of this area, if you want to know the area because I'm going to put two carpet over this thing, I don't want to buy zero square feet of carpet, but the integral, if I drew it right, it is certainly possible. So it could be true that the integral from zero to five of this function, after that, you come up with zero. That's because if you think about this definition, in this part of the rectangle, I have a negative height because the formula does not say take the absolute value, which is the distance from the graph to the axis, but just take the value of the form. You have to change the sign here. But if we want integral, then we think of this as negative. We think of this as positive. So integral, by this definition, without putting little absolute values, there is not actually an area. It's a signed area. Actually, I mean, that may seem like a defect, and it is a defect from the original design, but it's easily remedied. The problem is discover how much carpet I need. The integral is more than finding distances, a way of finding all sorts of things, and we can easily correct it by just taking absolute values. So it's not a defect that something with a positive area in the sense of carpet needed has a zero in calculating anything. It's just built into the problem that this is none of the negative because it's below the way we do it. It's negative. Some simple integral without, probably did, and on the homework assignment, you can calculate the integral without having to calculate the limit. For example, let's take a specific function of absolute value of x. Absolute value of x looks like this. Absolute value of x dx is, well, your clicker says 1a. If your answer is c, do not click the button marked 1. If your answer is c, click the button marked 3. Sorry. We should also say 1 slash c. Yes, it does. So do it by left. Anybody need more time? Okay, so I'm going to stop this now. Last chance. Here we go. Come on. You didn't answer when I said you need time. It seems that 14% of you think this is the right answer, but I won't. 12% of you think this is the right answer. 75% think that's the right answer. So most of you think that it's c, and most of you aren't right. It's unfortunate for those of you that think it's those. But if you just look at this picture, this is a question about area. It's 1 because the absolute value of 1 is 1. The width is 1. So this area is 1.5 above the axis. So in there, you could also then cut this out, lay it down there into 1 by 1 square. If the area of a 1 by 1 square is not 1, something is very wrong. So the answer is in fact 1. From 0 to 1, no. Negative 1 to 1 of x dx. The choices are a, hold 1 there. That person put it back. Good for you. The correct answer favored by 84% of you is in fact 0 because this stuff is plus, this stuff is minus from 1 from there to there. They cancel out and we get 0. We can use, we can break up integrals because their area can integrate from here to here and then integrate from here. It's the same thing as the integral from a to b because we're saying, okay, we have this function times pi f, 3 pi, so we must be clear. 0 because it has no width. It changes the sign. We go backwards. So once in a while, do some of this going integral from a to b. Everything is negative as the integral from b to b.