 Hi, I'm Zor. Welcome to a new Zor education. Today I would like to start talking about definite integrals. We have just completed, at least theoretically completed, the part of the course which is related to indefinite integrals. And now we're talking about definite integrals. Well, they basically sound very close to each other, right? Well, they are, but the way how I would like to introduce the concept of definite integral will be completely unrelated to indefinite integrals. Now, well, this lecture, as all previous and future ones, is part of the course of advanced mathematics on Unisor.com. I do suggest you to watch this lecture from this website rather than from YouTube or anywhere else, because the site itself has very detailed explanation, like a textbook basically for each lecture. And in addition, if you are a registered student, then you can take exams, for instance, just for your own satisfaction. The site is free, so no financial strings attached. So Unisor.com, that's where I recommend you to watch this lecture. Now, I will not talk about definite integrals today, but I will introduce you to a problem, solution of which will lead us eventually to a concept of definite integral. So, what is the problem? The problem is very simple. Let's consider you have a function which is defined on some segment, real numbers from A to B. It's defined on a segment, including both ends, and let's assume that the function is smooth enough, which means it's continuous, and it has the first derivative, which is continuous, maybe the second derivative. Well, I don't want to go into the detail of what exactly the smoothness means, but if I will use the first derivative, and I'm saying that the first derivative is continuous and has certain limits, like maximum and minimum, well, it means that the function is at least sufficiently differentiable to have this first derivative and its continuous. I will use these properties without probably even mentioning them. For instance, one of the property, if it's a continuous function, it reaches the maximum and minimum on this particular segment. If its first derivative is continuous, then the first derivative is reaching maximum and minimum. So, if I will use the maximum, it means the function is continuous and smooth enough. So, we have a smooth function, f at x, and what I'm interested in is, I'm interested in area under this curve. From A to B on left and right, from the graph to the x-axis from top to bottom, and let's assume that this is greater or equal than zero, so it doesn't really cross the x-axis, right? So, that's my function, and I need this area. Well, my first question is, what is the area? It's not such a simple question, because you think that you understand what this actual area is. You intuitively understand it, as well as everybody else. But, if you want to really define it mathematically, rigorously, you really have to do some work. All we know is, by definition, the area of a rectangle, let's say, it has widths and it has heights, is, by definition, widths times height. This is my area, and this is a definition, right? So, there is no problem with that. And that's the only thing which I know about the area. Somehow, I have to define first what this area is about, and well, come up with certain process, how I can calculate this area. So, first of all, the definition. All right, so let's just think about the definition. If you remember, we were addressing the area of a circle, for instance. We basically divided this circle into certain more simple areas, like triangles or whatever, and approximated. And we have introduced the whole process of approximation of the area of a circle with certain known areas of triangles inscribed into it, or whatever we have, or polygons or whatever. So, I would like to approach this problem in exactly similar fashion. I would like to build a process as a result of which we will get some number which we can say, okay, this number is the area of this particular part of the plane, bounded by graph, x-axis, and two vertical lines. So, what is the process? Okay, now we're talking. Now we have to really go into the details of this, and let's introduce the process. As in case of a circle, I will first break this particular area into certain pieces, and I will approximate the area of each piece. Here is how. So, let's break it into certain pieces, like this. By vertical lines. So, A is our x0, this is x1, this is x2, this is xn-1, and B is xn. So, we have n pieces, which we have divided the segment AB. Okay. Now, within each piece, within each, let's call it piece, within each piece, the function fattacks is reaching its local maximum and local minimum. So, let's talk about local minimum, for instance. Well, local minimum on this piece is this one. So, I will build this rectangle. Local minimum on this piece is this one. This is the point. On this, the point on the left is the point of a minimum, and here and here. So, what I'm saying right now is that instead of measuring the area under the curve, which I don't know actually what it is, I can say that I know the area of these rectangles, and I can add them up, and I get some approximate value for something which I intuitively understand must be the area of this part of the plane under the curve, right? So, the area under the curve can be approximated by some of these rectangles. Okay. What if I will choose different partitioning, different points? Well, I will have a different number. So, obviously, I have a little problem here, because I cannot really define my process exactly how it's supposed to be done, performed, to get some kind of a number. So, I can say that, okay, let's divide it into n equal parts, and I will take the minimum of the function f at x on each part, build a rectangle of the height equal to the minimum, and I'm saying, okay, this is approximation. All right, fine. But then what? It's just approximation. It's not the exact area under the curve, right? Well, I can say the following, actually, that if I have an approximation based on this particular partitioning, and I will choose to, let's say, break one particular interval in two. Let me just make a little bit bigger picture. So, let's consider this is my xi minus one, and this is xi. Just one interval. I have blown it up, all right? Now, the minimum of this function looks like it's in this particular point, and this rectangle is my approximation. What if I will introduce another point here, x prime? And I will break this interval into two. So, I will build this, and I will have to have minimum on this rectangle and minimum on this. Now, obviously, if my initial minimum belongs to this piece from x minus one to x prime, then this particular part will be the same. But in this case, minimum will be different, will be this. So, obviously, minimum on this part will be greater than the minimum on this part if my initial minimum for the whole interval is here, right? Which means that some of these two rectangles, this one and this one, is greater than my initial rectangle. Which means that if I will make a finer partitioning, if I will introduce additional points into existing partitioning, then some of my rectangles will increase, because the minimum on the new one would be one of these new ones, will be greater than the minimum for the whole interval which was used before. So, I have two rectangles, one of them will be of the same height as before, and one of them will be higher, because I am always choosing the minimum. So, minimum on this is obviously less than minimum on this. So, as I am introducing new points into my partitioning, this particular area of the sum or sum of the areas of rectangles is increasing. That's a very important point. Now, let's approach it differently. Instead of choosing the minimum, I will choose the maximum. What happens? Well, in this case, maximum is here, in this case, maximum is here, in this case, maximum is here, here and here. So, all my rectangles become higher. Now, if before, all rectangles lie completely below the curve, because I am always using the minimum of the function on any particular interval. Now, I am taking the maximum, which means all rectangles will be completely outside. So, in the first case, my curve was above the rectangles. In the second case, when I am choosing the maximum, all the rectangles will be above my curve. Now, what happens if I will introduce additional points? Let's think about this way. So, I am not using this rectangle or this rectangle. I am choosing the maximum now. So, what's the maximum overall? It seems to be somewhere here, right? So, my initial rectangle will be here. Now, instead of that, I am introducing a new point X prime, dividing my interval into two parts. Now, this maximum lies in this piece, which means I will have this rectangle here, and a little lower, that seems to be the maximum on this interval, will be here. So, my point here is that if I introduce additional points into this set of rectangles, where I am using the maximum of the function, initial point will reduce the sum of the areas of all rectangles. Now, this is very important. The minimum function on each interval will result in the sum of rectangles, which is increasing as I am introducing new points. If I choose the maximum, the sum of rectangles, sum of areas of rectangles, will go down as I introduce different points. Now, let's recall the properties of sequences. Remember, if my sequence is monotonic and bounded from above, let's say it's monotonically increasing and bounded from above, it has a limit. Similarly, if my sequence is monotonically decreasing and is bounded from below, then it has a limit. So, what does it mean? It means the following. Let's call minimum of the function on I's interval mi and maximum mi. So, the sum of rectangles, the sum of rectangles which use the height, the minimum on each one is equal to sigma mi xi minus xi minus one. Or sometimes we use delta xi. Delta xi is basically xi minus x minus one, the increment of the lengths of the interval. Now, let's call it lowercase sn. That's the sum of areas of rectangles. Mi would be the height and xi minus i minus one would be the width. And its multiplication is the area of rectangle. Now, I'm summing it from i equals one to n. And that's how I get my sum of rectangles which are below the curve. Now, another, let's say capital Sn would be very similar one. Now, I will use the uppercase mi which is the maximum of the function f at x on the I's interval. And if I multiply it by the widths of the I's interval, I will have the sum of the areas of rectangles which are above my curve. So, what do I know about lowercase sn and uppercase sn? These are sequences as n is increasing if I'm introducing more and more points to divide my initial segment into n parts where n is growing. Then lowercase sn is increasing to some limit. I know that there is a limit because it's monotonically increasing and it's bounded from above by let's say anything like this. I mean the corresponding sn is definitely greater than sn and then it's only growing. Or if you wish you can just draw a rectangle like this and obviously this would be significantly higher than any limit of sn. So sn is bounded from above and monotonically increasing and therefore it has a limit let's call it sn infinity. So sn goes to infinity. There is some kind of a limit. Now, what's important is here that whenever I'm introducing new points I'm actually making smaller each particular interval. So let's agree that whenever I'm introducing new points I'm always introducing it in such a way that the largest interval goes down to zero the widths of the largest interval. That's very important because I can actually divide the segment into n parts and then introduce all my additional division points somewhere between this and this leaving this alone. Now that's not what I'm talking about. The process I'm talking about is the process of introducing new partitioning points in such a way that the length of the largest interval always goes to zero. Okay, now what happens with uppercase sn? Well that's monotonically decreasing with introduction of new points as we were seeing before and again it's obviously limited from below, well at least by zero right because it's all positive numbers which means it also has some limit. Capital S infinity. So what's interesting is that as I increase number of points here those rectangles which are below are tighter and tighter approach my curve which means that the limit which it's basically converges to might be a good candidate for the area under the curve. Very similarly if I will consider the taller rectangle they are above the curve but as the number of division points is increasing they are tighter and tighter encompass the curve itself which means their limit also might be a good candidate for the area under the curve. Well it would be great if these two limits are the same, right? So if they are the same that's actually would be a very big deal because it means that I can take any partitioning and I can take either the minimum point or the maximum point or anything in between actually as a value for the height of the rectangle anything in between, I mean as a value of the function in between any two points which define my interval. So that gives me a lot of flexibility and it gives my definition certain rigidity, certain affinity if you wish because no matter how I calculate using minimums, using maximums, using any in between points etc. I will still have the same answer the limit which is this or this if they are the same. Well these are the same and I would like to prove it. So if I will be able to prove these two numbers are the same under condition of increasing the number of partition points and the length of the widest interval being infinitesimal variable going down to zero. If I will prove this then the whole definition as area under the curve is equal to the limit of the sum like this provided that the largest delta Xi is diminishing to zero as n increasing to infinity then that would be a nice and valid logical definition without any kind of certain very fluid kind of explanation etc. No matter how you calculate it you will have the same number. Alright so let's concentrate on proving that the difference between these two is actually going to zero as my n increasing to infinity and the delta Xi, maximum of delta Xi among all the delta Xi's goes to zero. And that's actually a relatively simple thing. Here is what I will remind you. Okay I would like to remind you the Lagrange theorem from the derivative. If you have a function and you have this is f of b, this is f of a, this is f of a, this is f of a, this is f of b and this is the function f of x. Then the difference between f of b minus f of a is equal to some point, there is a point in between a and b and there is a derivative in this point such that difference between this and this which is actually the length of this is equal to b minus a times basically tangent of this angle. So if this is the chord and this tangential line is parallel to the chord then these angles are the same, right? The same as this one. So it's basically a right triangle, right? This catatouche is equal to this catatouche times tangent of this angle, right? And tangent of this angle is the same as tangent of this line with an x which is a derivative. Remember, derivative is a tangent of this particular angle. So I will use this particular theorem. Now, why? Now let's do this. Capital Sn minus lowercase Sn is equal to sum mi minus mi times delta xi, right? Where mi is a maximum and lowercase mi is a minimum on the interval from xi minus 1 to xi. So let me again increase the scale of this picture and consider only one particular interval. So this is my function. This is my xi minus 1. This is my xi. This is the point of a maximum. Let's call it xi. This is the point of my minimum. So that's the same as, let's call it eta i. So xi i is the point of a maximum and eta i is the point of a minimum of the function f at x on the i's interval. So my capital mi is this. This length. My lowercase mi is this. Now the difference between them according to this theorem is equal to, so let's find a third point. Let's call it zeta i which is in between xi i and eta i. And if I will take the tangential line or derivative if you wish in this particular point, then the difference between capital mi and lowercase mi, which means this, is equal to the difference between these two. So let me do it again. Mi minus mi. Is equal to xi minus eta i. I'm using absolute value not to be bothered. Times tangent of this, which is actually f derivative at point zeta i. Right? Which is less than or equal to. Now instead of this, now both xi and eta i inside this interval. So they are definitely less than delta xi. Right? Difference between them less than the difference between the edges of this interval. Now this is a derivative of the function at some point inside this interval. Now I have assumed that the function is smooth enough, which means it's differentiable and it's derivative is continuous. And if derivative is continuous on any particular segment, then it reaches its maximum and the minimum. So let's say there is some number, some constant, which is the maximum of the derivative of the absolute value of, I have to put absolute value here. So times k. Where k is a constant, which is a maximum of derivative of the function f of x on the whole segment from a to b. Again, if the function is smooth, it's continuous and its derivative is continuous, which means derivative is reaching its maximum and minimum. So there is a maximum. Just consider that the maximum is a constant, it's some kind of a constant k. So all I have to do is basically this. Right? Now, I have also said that my process of increasing the number of points is such that the maximum interval is shortening and shortening down to the zero in the limit, converges to zero. The width of the largest interval is infinitesimal. So let's consider there is something which we call a largest interval in any partitioning of our segment a, b. But obviously it depends on the number n. So on each step when I'm introducing new points, there is some interval which is the largest. Right? So therefore, any interval would be less than the largest interval. Let's call it, let's say delta n times k. So delta n is a variable. Now it's at the constant. It's a variable and it's changing with increasing n. Delta n is actually the largest widths among all the interval on the nth step of my partitioning of my segment. Okay, that's all I need. Now, considering this, I can say that this particular difference between s and s is less than, okay, mi minus mi, capital minus, I will use this. So it's less than sigma delta n k times delta xi. Now these are basically independent of summation. This is i. So I can bring them outside of the sigma. Now what is the sum of delta xi? Well, that's x first minus x zero plus x two minus x one plus x three minus x two, etc. Which means it's basically sum of all these intervals together, which is the whole length of my segment a, b. So it's k times delta n times b minus a. So what can we say right now? If my widest interval is getting shorter and shorter, it shrinks as I'm increasing the number of partitioning points here. Then this is infinitesimal, so the whole thing is infinitesimal, this is the constant, this is the constant. Which means the whole thing is going to zero. So this difference between these two converges to zero. Well, which means that whatever the limit of one of them is exactly the same as the limit of another one. They are getting together. Lowercase Sn is increasing, uppercase Sn is decreasing, and the difference between them is infinitesimal, which means the limit is supposed to be the same. If limit is different, obviously the difference would be not infinitesimal. And this is the end basically of the whole theory behind area under the curve. Because now I can say that to get the length, to get the area under the curve, I have to split the segment where I would like to determine the area into n parts. To calculate the sum of the area of rectangles, where the height of the rectangle can be either the minimum, or the maximum, or the left boundary, or the right boundary on each interval. So I can actually have, for instance, I have just two partitioning, partitioning in two different intervals. As the height of the rectangle, I can take minimum, or I can take maximum, or I can take left boundary, or I can take the right boundary, which is here. But the result of summation as the number of intervals is increasing, and the widths of the widest interval goes to zero, then the result, the limit will be exactly the same. And that limit, limit of whatever the number we are calculating as sum of the area of rectangles, whatever the limit is, we will call by definition the area under the curve. So, whatever we were doing right now, we did not calculate actually the area under rectangle, under the curve. Actually, I will do it in the next lecture. What we have done, we have defined, this is just definition, and I was trying to prove that the definition makes sense. Because when I define something, area under the curve, it's supposed, well, first of all, it's supposed to exist. Does it exist? Yes, because I defined it as limits, and I have proven that the limits do exist. And secondly, it should be unique, regardless of how I obtain these limits, regardless of the process, as long as the process is well defined. And the process is well defined, just as long as my partitioning is finer and finer, as long as the widest interval is converging to zero, my limit will be the same, regardless of how I choose the height of the corresponding rectangles on every step. Okay, so this is the end of this particular lecture. Again, my purpose was to define relatively rigorously what is the area under the curve. Actually, I shortcut something. You see, I have defined this and have proven that it makes sense for smooth functions. Basically, in certain cases, the theorem can be proven actually with much weaker conditions. For instance, if I will choose my curve like this, between this and this AB, I mean, obviously you understand that it does make sense to talk about the area. Now, what's my problem? The function is not differentiable at this particular point. So there is a derivative on the left, there is a derivative on the right of that point, and at that particular point, derivative does not exist, function is not differentiable, which means for this type of function, my proof actually is not good. However, in my defense, I have to tell that the proof, for instance, for continuous functions without regarding to their differentiability, for instance, is a little bit more difficult. And my purpose was not really to absolutely rigorously, in the most general case, define what the area is. My purpose was to explain you that the area can be obtained as the result of the process, as a limit value of certain steps which you are executing. And the fact that I have proven this for a little bit narrower set of functions, so this function is not really part of this set. Again, it's more illustrational than for the real purpose. The purpose of this lecture is for you to understand how to define the proper definition, how to make up the proper definition of the area under the curve, and how I proved this is a good and valid definition is just quite frankly beside the point. All right, so I think what I will do in the next lecture, I will try to use this particular definition in a couple of cases and to demonstrate how we can calculate the area under the curve using this particular definition, which means having certain process which results in some limit which we will take and we will get some result. Okay, that's it for today. I do recommend you to read all the notes for this lecture on unison.com. That's it. Thanks and good luck.