 The idea in polar graphs, we have something like the radius is a function of the angle. So, for example, something like this, r equals 3 sine theta gives us a circle because we start off with theta is zero looking this way. The radius is zero and because the graph of the sine increases from here to here, we go out all the way to here where the radius is three. And then as we go in the second quadrant, the sine decreases, we get something like that. So, it would be silly to ask for, well, we could do this if you wanted. But to, I mean, we know the area of this, it's 9 pi squared. So, let's also look at, let's say another part of this guy, 1 plus sine theta. This is also one that we did. This is a, well, we can just think about it. When we're looking at the zero degree angle, the radius is one. So, we start, since this is three, I guess one is about here. So, we start here somewhere, exaggerate and come out here. And then the sine, again, increases up to one. So, that means that as we go up here, the sine will increase and we will cross here when we're looking straight up at a radius of about two. And then the sine will decrease again down to, when we're looking this way, when the sine is zero again at pi. Are people following this, okay? Yeah, okay. And then it will continue to decrease. So, since it's one plus sine, the sine will be negative. So, then we'll get radii less than one until at when the sine is negative one. And so, we'll come in and we'll get cartiway like that. So, this graph, well, it's hard to tell which one I'm pointing at. That one is one plus sine theta and this one is three sine theta. And let's find the area, well, my picture's a little off of this crescent shape. This works just like when we're finding rectangular area except that we're thinking in polar, we need to know, my picture's a little distorted. Let me distort it more so at least it's symmetric. Oh, it's even more wrong, but at least it's symmetric. So, what do we need to know in order to find this? Where they intersect. We need to know this point and this point and we're going to integrate our angles from here to here. So, we're going to integrate over that range of theta's. So, how do we figure out where they intersect? We solve where they cross and don't need this line here. So, obviously, what they cross, the radii we need to say. So, we need three sine theta equals one plus two, no, one plus sine theta or two sine theta is one or the sine of theta is a half. So, where is the sine of half? This is one of those things we should know here, 30 degrees or pi over six, so since we want calculus to work properly, we need to work in radians. So, this angle here is pi over six and by symmetry, this angle here is pi minus pi over six or five pi over six. Or 150 degrees or whatever you want to call it. So, that's what we want to do. And so, now the area is definitely a pi d theta, okay, sorry. Right? Because what are we doing? We're finding this area and then we're throwing away this area. So, it will be another way to write this. It's the integral of pi r squared, so this is pi r squared but we don't, there is no pi, you're right. Oh, it's this stupid thing, I'm so mad. So, one half, pi a half, what's the difference? Okay. So, it's pi r squared but we don't have the whole, we don't have the whole circle. So, if we have the whole circle of the pi r squared, but we only have a d theta of the circle, so we get a half because the pi is cancelled. And then we get the same thing for the integral. Okay, good thing I abandoned this quicker question because it was just completely wrong. And then we go from the start, which is pi over six, to the end, which is five pi over six. Or if you prefer, you can go from pi over six to pi over two, you may be done in double because it's symmetric with respect to the y-axis, yeah. Okay, this is the outer one, this is the inner one. We're finding the area between two curves, so we find the area from the outer one back to zero and subtract off the area from the inner one back to zero. That will give us that piece, okay? You not see that? I mean, if you don't see that, tell me, I'll give you that. I know, and why is it a half? I'll say why it's a half. I answered the second question first. So, the first question is why is this part that? And this is because this is the r squared from the outer circle and this is the r squared from the inner circle. Okay? Okay, so why isn't it pi r squared? It would be pi r squared if instead of taking a little piece of thickness d theta, we have a whole circle. We don't have a whole circle, right? We just have a little angle d theta. So, the angle d theta is being one degree. If we had one degree, well pi r squared is the whole circle, so d theta would have to be one over 360. So, we have to divide by one over 360, but it's not one degree. It's a d theta of a circle. So, we have how much of the circle? The whole circle is two pi radians. We don't have the whole circle. We have only d theta of the circle, but of course, we get a pi from the pi r squared. And then the pi's cancel, the pi's cancel leaving us with just a half. So, I went through this on Friday or whatever day it was, but maybe you missed it or you got it or you're not here or whatever. Okay, so does everybody agree with me that this is the right integral? Anyone confused about why this is the integral? Can you point your finger at what portion of this integral can be used? You don't understand why? Okay, if I wanted a whole circle, it would be pi r squared. Right? I don't have a whole circle. Instead of having the entire piece of pi, which would have radians pi r squared, I only have a thin little slice of angle d theta. I think that's exactly what it is. It's pi r squared divided by whatever section of the circle I have. The whole circle is 2 pi. I only have d theta of it. And but here's a pi, there's a pi, they cancel. So the whole circle is in 2 pi. If I were doing this in degrees, maybe it would be clearer that I would have pi r squared, then I got to multiply by 360, I mean d theta over 360. The problem is I can't do it in degrees because then calculus comes up with nasty constants. So that's why it's not pi r squared, it's one half. Okay. You wanted to know again why and where are you, there you are, why the shapes are what they are. Okay, so quick review, even though I don't want this to be polar forever because it's cold there. Okay, so if we have, say, the circle 1, r equals sine theta. So the radius is given by the sine of the angle we look in. If we look in angle 0, the radius is 0. If we look in angle pi over 6 or 30 degrees, the radius is one half. Oh, sorry, I wanted 3 sine theta. So the radius is 3 halves, so let's move it out first. As I go from here to there, in a nice smooth way in terms of the angle, the radius goes from here to there. So I'm going to get some kind of a curve like that because as I'm moving my angle up, the radius is increasing. The graph of the sine in rectangular coordinates looks like this, here's pi over 2. So as I go from here to here, the radius increases. Up here at 45 degrees, the radius is still increasing. Up here at 60 degrees, the radius continues to increase. So this curve is just going to continue to grow. But here, it grows quickly, then more slowly. So it will tend to slow down. And here, at pi over 2 degrees when we're looking straight up, the radius has reached its maximum because the sine has grown as much as it's going to grow. And then it starts falling. And it falls in an exactly symmetric way to the way that it grew. So the curve over here will be exactly the mirror image of the curve over there. It will be some curve like this. Now maybe this isn't a circle. For the purposes of this question, it doesn't matter. It is a circle. I think you have a homework problem to show that it's actually a circle. But it's a circle. Okay? So that's one of the curves. The other curves, r equals 1 plus the sine of theta. Again, we start at, well, the sine of theta is 0. So we start when we're looking this way at a radius of 1. Some distance from 0. Now as theta goes up to pi over 2, as we go from looking this way, it will be this way, the radius grows up to 2. And it grows in a nice smooth way like this, but it starts out here. So it grows something kind of sort of like a circle. It's not really a circle. It looks kind of like a circle. And then when we go from pi over 2 back down to pi, the radius begins to shrink. And again, it looks sort of like the top part of a circle. It's not actually a circle. It looks kind of sort of like an upper portion of a circle. Now things get a little more complicated down here. We're here at radius 1. My picture is a very similar, but we're here at radius 1. And when for the angles in here, the sine is negative. So our radii will be less than 1. And as we go to looking straight down at 3 pi over 2, the radius goes all the way down to 0 because the sine goes to negative 1. So this thing continues to shrink and it comes in to an angle of 0. You can check that this is vertical by taking the derivative and seeing that the radial derivative here gives you something rude. And then it grows in the same sort of way that it shrank before that. Now this is a sloppy picture. It's a cartiloy. It looks like it is what it is. You can plot it on a computer and see that that's what it really looks like. I did some of that on the front. Okay. I want to move along from polar coordinates because I'm really kind of higher than them. And we need to move on to the next portion of the course. I thought this would be a nice 10 minute exercise and clearly it wasn't. So it's worth your while to practice playing with these things. This is only a small portion of the course but it's something that you should understand. But now we move on to the next big section of the course except we actually move on into it in a slightly different way. So in some sense this course has a hidden agenda. You think that this is a course about integration, maybe infinite series, maybe differential equations. That's true. But it also has a hidden agenda of giving us another collection of functions to work with that don't fit into the standard form of sines, cosines. That's potentials and logarithms. So for example, this is, so I'm making a little divergence from actually starting. If I do something like f of x equals the integral from 0 to x of sine t over t, well, yeah, sure, that's fine. This is a perfectly good function. It does not have a nice formula any better than this one that I wrote. Another one, let's not call it Earth because I don't want to divide by 1 over 2, let's just call it capital E. Another one, 0 to t, I switch my t's and my x's into the minus x square over 2 dx. This is a very perfectly good and useful function. So in some sense, the first part of the course in addition to figuring out how to calculate integrals is to realize that we can write additional kinds of functions in terms of integrals. So this sort of expands your repertoire of what kind of functions you can deal with. Derivatives don't give us new things, but integrals do. In the same sense that multiplications we start with whole numbers does not give us new numbers, but division does. We have 3 times 4, we get 12, this gives, we take two whole numbers, we get a new one, but this is still a whole number that we already have. But if we take 4 divided by 3, we get something that is different from what we started with. These inverse operations sometimes give us extra things. So derivatives are, taking derivatives is kind of like multiplying and integrating is kind of like dividing. You can start off with the old functions that we had take their derivatives and we get no new functions. But if we start with functions and we integrate them, we get new functions. So this wrap sort of is one way of getting new functions. Another way of getting new functions is by adding lots of them together. So we've already seen things. So if we add together say x plus x squared, this gives us a more complex divided by 2. This gives us a starting line. This gives us slightly more complicated than we had before. And then, if I just keep going, so far I still have a polynomial. But I can imagine that I do this forever. This gives me an increasingly more complicated object. And the limit of this is something that is not a polynomial. We will explore this in great detail later. Okay? So this kind of thing, which we can write this. And I know that the sigma notation that we had before confused people. Well, now we're going to use it a lot. So let's say n equals 0 to infinity of x to the n over n factorial. This is just a short way to write this. First I let n be 0, 0 factorial is 1, so x to the 0 is 1. Then I let n be 1, x to the 1 is 1 divided by 1 factorial is just x. Then I let n be 2, x to the 2 divided by 2 factorial is this. And I let n be 3, x to the 3, 3 factorial is 6. And I let n be 4, 4, 1, and so on. And I just do this. And I do this forever because this goes to infinity. So one of the things that we'll be doing for the next couple of weeks is working up towards understanding what this means. This is actually a function we know. This is e to the x. But to see that it's e to the x is perhaps requires a lot of ground work. So for the next couple of weeks we're going to lay some ground work to work up towards having additional kinds of functions that are described as not as integrals of functions that we know, but as infinite sums of functions that we know. But we have to work on making sense of infinite sums. And infinite sums in general. And then infinite sums of functions. Okay? This is a very powerful technique. And it's very useful in a lot of applications of mathematics and science to allow us to extend this way and understand these kinds of functions that seem like they kind of don't make sense. We can think of them as approximations if we stop somewhere. Or we can think of them as objects that are much more common. So this is my little introduction to the next probably four weeks of the class, maybe three. Probably four. Okay. But before we can actually study this kind of complicated object, which is not really there. But before we can study this kind of object we need to build up some tools. Just like before when you started calculus, before you could really start talking about derivatives, you had to build up some of the notion of limit. So here we have to build up what this possible we need. Okay? So there's a little side trick that we need to take, which is not really a side trick. And it confuses students because we talk about one thing that we immediately start talking about something that seems to be different. This is the notion of an infinite sequence. Now our main focus here is on something like this, which is called an infinite series. I will try to use the word always sum to emphasize that it's stuff added together. This is a special kind of infinite sequence. A sequence, so one of the problems with this word series versus sequence is in English, we kind of use them interchangeably. Right? We say this was an unfortunate series of events or this was an unfortunate sequence of events. They mean the same thing in English. But in mathematics they mean different things. A sequence is just a list of numbers. So something like I got tired. This is an infinite sequence. We call this guy the first element. This one's the second, this is the third, blah, blah, blah, blah, pretty straightforward. You could have another one that looks like one, one half, one third, one fourth, one fifth, et cetera. Again, pretty boring. We could have one that looks like three, one, four, one, five, two, seven, et cetera. There's another one. So we don't know what the next digit is, one. Does anyone even have a clue where this sequence came from? It's the digits of pi, okay? 3.1415, et cetera, 9, 9, 2, 7, I think the next one's 1, but I'm not sure. I think this one's 1. Anyway, it doesn't matter. This is the sequence where the nth digit is the digit of pi. We often would denote this using curly brace notations and some index like an n. When we say where it starts, in this case let's start it at 1 to infinity. So this notation means the list that starts with a1, then a2, then a3, then a4, and goes on forever. So this notation here is just a shorthand way of writing this long list. This long list is perhaps somewhat complicated. So for example, this one is easy. This is the sequence 1 over n from n equals 1 to infinity. And this one is just the sequence n, n equals 1 to infinity. Okay? So the sequence is just a list of numbers in order and we can talk about the 75th number or the 2018th number or the nth number. And what we're mostly concerned with regarding sequences is whether they converge or not. So on the board here, this is the only one that converges. So what we would like to know and what's useful about them is, so does this sequence tend to some number? That is, so what I want to say here is if there is an L, using an L because it stands for limit, so if there's some number L, so that the difference between the nth element and L is as small as we like for all n, let's say little n, beyond some form, then we say converges to L. So the sequence of n, and we often overload notation that we already have, we already have the notation and we lift this up and I'll put it back down again. The limit is n goes to infinity. So this is a concept that you've already encountered but it's not a function, it's just a list of numbers. We use the same words to describe this behavior because it behaves just the same way. If this list of numbers is closer and closer to some other number, then we say that the limit of the sequence as we go to infinity is L. Unlike limits of functions, it doesn't make sense to talk about limits as n goes to anything except infinity. Sequences only go one way and this statement is just saying that this function goes to L as n gets big. It can't go anywhere else because, I mean, n1-has doesn't make any sense. We can't have a limit of the half term, okay? So sequences are pretty simple. They're really just the same as limits that we had before and they are sort of the underlying tool that we will use to study series. So we're not going to talk a whole lot about sequences. We'll talk a little bit about sequences. So because you already know about limits, the usual limit laws apply. So if we multiply two sequences together term by term, then the limit is the same. And they both converge and the limit is the same. If there is some function so that when we plug in whole numbers, we get our sequence then if we take the limit as x goes to infinity of the half of x, if this is some number, our sequence A n will converge to that number. It has no choice but the reverse is not necessarily true. If the kind of limit that you already knew about as we go to infinity is something, then the sequence goes there but the reverse is not true. So for example, what might be true might not be true. So for example, suppose I let my sequence, we can start at zero this time, be the sine of n times pi. Let's make it 2a. So it's not true that as x goes to infinity of the sine of x does not exist because the sine goes up and down and up and down. And there's no limit here because it's always going up and down. But this is not looking at all the x's. It's only looking at the ones every 2 pi. So if I start listing these out, when n is zero, sine of zero is zero. When n is 1, sine of 2 pi is zero. When n is 2, sine of 4 pi is zero. When n is 3, sine of 6 pi is zero. This is all the zero. This one is just looking at it here, here, here, here, every 2 pi. So this is sort of like, I don't know, I ask people this and I discover that they don't do this. When I always did this when I was a little kid, you're driving down the freeway, the LIE and you look out the window of the car next to you and you blink your eyes just right, it doesn't look like the spokes on the wheel move. Of course, has anyone done this? Am I the only person who's ever done this? That's amazing. Okay, he did it, good for you. So imagine that you're sitting, well, you certainly see this when they, when they, you're watching a movie and they film a TV set. And there's this strike that seems to move across the TV set. That's because the film is taking a photograph or a still image of the TV set every so often and it doesn't line up with, the TV's also flashing a light, they don't quite line up and you see a little strike that's in between. The same business here. If we just look every 2 pi, the sine is always zero. But if we look at the whole thing, it's wiggled. Okay, next time you're riding in a car, don't do it while you're driving together. Next time you're riding in a car, look at the car next to you with the link your eyes and see if you can see the spokes and the hubcaps and they don't move. Maybe they just ride. Okay, now there's a couple of little useful techniques that we can do with series and I hope I can get through them in 6 minutes. So one thing that we can talk about, just like with functions. So the sequence AN is called increasing if, surprise, surprise, the next term is always bigger than the previous term. Sometimes you can allow it to be equal and then it's called non-decreasing or just to emphasize that it's strictly increasing, sometimes we use the word strict. So let's call it strictly just to emphasize. And it's called decreasing, again strictly, if the next term is smaller than the previous term. Okay, and it's easy to check this sort of thing by just messing around with algebra or if it corresponds to a function you can take the derivative and see if the derivative is positive or negative and so on. But what's useful about this, you can just think about, okay, so, okay, so, and then lastly a sequence it's called bounded if there's some number that it's never bigger than n or let's say it's always between and minus n. If there's some number that it's between. So for example, this sequence, well this one is stupid, but this sequence sine of n is bounded. We can let it be equal because the sine is never bigger than plus one and never smaller than minus one. Yeah, maximum, Michael, whatever you want it to stand for, it's just the letter that I chose to stand for some number. There's a number n that traps every element between it. So it never gets bigger, so like the sine never gets bigger than one, never gets smaller than minus one and this is a bounded sequence. Notice that this does not converge. I don't know what the sine of one is, I don't know what the sine of two is, but it's not a nice number and it just sort of bounces all the way. Okay, so the point I want to make, if we think about these sequences, say in terms of a picture, you draw a graph, what happens if the sequence is bounded and increasing? So if it's bounded, it means that there's some line that can never get above and I just start and it does its thing, but I can never get above that line and if it's increasing, then I can never go back down even. So if I want to make this wiggle a lot, every time I go up, I can't go back down so I'm kind of forced to converge and I don't care what it's bounded by but it can't wiggle, so it can't escape off to infinity because the bound stops it and it can't wiggle around because every time it goes up, it can't back up, it can't wiggle higher, so it always goes higher and it can't get above a certain height, so it has to converge, so this is maybe a not obvious thing, this is a little bit hard to include explicitly but it's very useful. Let me say one more thing, say really don't want to talk much about sequences. One other thing that we can define sequences, I define sequences mostly by formula but we can also define them what's called recursively, so those of you that do computer science who've already encountered, you can probably encounter it in recursively a lot, so we can define a sequence recursively, this makes it a lot trickier but it actually arises this way naturally, so for example, we might say the first term is one and then, well let's pick it up too, and then the next term for any n to get the next term, I take the previous term and I do something to it, something like that. Now here, this doesn't tell me exactly what A47 is but I can figure it out because I start with 2 and then to get the next term, I take 2 and I divide it by 3 and I add it to 1 and then to get the next term, I take this, so this is 5 thirds and I divide it by 3 that gives me 5 ninths and I add it to 1 and this is 14 ninths and then I take the next term and I divide it by 3 and I add it to 1, so I can generate the sequence by this relation but write down exactly what the hundredth term is, is a little bit tricky that sometimes you can't do it at all in this case you can't, let me stop there.