 Hi, I'm Zor. Welcome to a new Zor education. Well, I think it's time to define what is a definite integral and talk about its properties. Basically, we have prepared with all the... I think I had three different lectures dedicated to different examples of certain needs in mathematics and physics where we approach the same thing basically using the same methodology and came up with very, very similar formulas. And that's the justification for introducing just these particular kinds of calculations, kind of formula, define it, and investigate its properties. So, definition of definite integral. We were dealing in all these previous problems which I have suggested with basically the same thing. If you have some kind of function, smooth function, continuous, differentiable maybe, defined on segment AD, and we are talking about real functions. So, this is a real argument and real function. And then we have introduced a process. On each step what we do, we divide our segment AD into different intervals. I'm using time, but it's actually x in this case. So, x0, x1, etc., xn. So, this is a and this is b. And you have n intervals in between. Now, on each step we calculate a sum. And then we do this process of making our intervals smaller and smaller by increasing the number of intervals and making sure that the maximum interval, the widest interval is still converging by lengths to zero. So, we introduce the limit. So, I will put n as it is converging to infinity and maximum of delta xi goes to zero. Now, this is usually abbreviated as delta xi. Now, what's very important, and I actually talked about this in the first lecture when we were talking about area under curve. We talked about that no matter how we divide our segment AB into different intervals, you can divide into equal intervals. Maybe not exactly equal. Maybe every odd interval is equal to another odd interval and even intervals were equal by themselves. Whatever the way, as long as the maximum interval, maximum by lengths interval is becoming shorter and shorter down to zero, which obviously necessitates that the number of intervals should be converging to infinity. So, as long as this condition is satisfied, the limit exists for our function f. And it's unique no matter how you approach this particular process. So, for example, what you can do is divide this interval AB by half on the first step. On the second step, you divide each half by half. So, basically getting four different intervals. Then, again, divide by half. All intervals you will have eight intervals. So, obviously, our condition is satisfied. The largest interval is shrinking to zero. The number of intervals, obviously, is going to infinity. And the particular calculations in limit will actually result in some kind of a number for each function f we choose, relatively smooth function f. We are talking about smooth function. Now, somebody else might decide it differently. Okay, I will define it in three different pieces this interval, equal pieces. And then each one-third I divide again in three pieces. So, I will have nine intervals. And then 27, et cetera. Well, it's a different process, but in the limit result will be exactly the same. So, existence of this limit and its uniqueness allows us to say, okay, this is some entity, some number, if you wish, which depends only on function f and the segment where it is defined. And that's why I can call this thing an integral, a definite integral from function f at x from a to b. And this is the symbolics of this. The sign of integral exactly the same, by the way, as for indefinite integrals. And we will talk about why we have the same word integral used in both cases. One is called indefinite integral and another is called, this one is called definite integral. They seem to have absolutely nothing in common, right? So, if you remember indefinite integral, it's basically anti-derivative. It's the new function, derivative of which is equal to original function, right? And this has something to do with these sums, which doesn't seem to be related in any way. And yet, we are using the same symbol, just put a couple of limits, the lower limit and the upper limit of integration. Okay, so we will touch this particular topic in the future. Right now, you can just consider this to be just a symbol, symbol which basically tells us the way how we come up with this thing. This is the way how we come up. We think about certain process of making our interval, our segment AB, consisting of intervals of smaller and smaller size. And in the limit, we have something. And that something is called integral of the function f of x in the limits from a to b. So, the definition is there. The definition makes sense because of uniqueness and existence of this particular limit on the left of this formula. Now, I actually calculated a couple of times the area under curve using this particular technique, if you remember. It's one of the, I think it was the second lecture about definite integrals. So, we can definitely do this. We can calculate our integral using this technique. But nobody is doing it right now this way. And the way how to properly address this. Again, I will talk about this in different time, different lecture will be. For today, I would like to concentrate only on the definition of this and elementary properties. So, let's talk about properties now. So, there are certain obvious properties of this integral. Property number one, if you will take integral of two functions sum together, it's equal to sum of their integrals. Now, why is that? Well, obviously, because if this is not just f, but f plus g, then I can multiply them separately by xi minus 1. Sum of sums will be, again, sum of sums and limit of sum is equal to sum of limit. So, basically, all the known properties of the summation and limits will be elementary used in this particular case to prove this formula. So, I'm not really talking about certain things which I consider to be really trivial. So, let's not just waste our precious time on this. Now, here is a little bit more interesting example. What if I will integrate not from a to b, but from b to a? Well, first of all, what does it mean? Well, it means basically that x0 is b and xn is a. So, if a is less than b, all these will be negative, right? So, the correct relation is this one, because these values will be the same, no matter how we go from left to right or from right to left. But if we go from right to left, we start from x0 equal to b and finish xn equal to a, then all these would be negative, right? Because xi would be smaller than xi minus 1. And that's why I have to put the minus here. So, if I change the limits of integration, it means I'm changing these delta xi's by sine, that's changed by sine, not only from positive to negative. And that's why integral from a to b and integral from b to a are related to each other just by this minus sign. This is something which you might not really expect. But it's very, again, it's a very simple property. Now, next property is the following. What if you integrate from a to a? Which means our segment, a, b, has zero lengths. Well, obviously, all xi's would be the same, they're all equal to each other. So, all these would be zero, some will be equal to zero and limits will be equal to zero, obviously, right? So, integration from a to a would result in zero. Next, again, is a little bit more trivial, I would say. If you have some kind of a multiplier, constant multiplier. What happens here? Well, if this is a constant multiplier, then obviously it presents itself here. You can always take it, factor it out from the sum. And obviously you can take it out from the limit because the limit of multiplier by some variable, you can put the multiplier in front of the limit. So, again, trivial properties of summation and going to a limit result in this. This is trivial. Next, next is interesting. Let's consider our function on a segment from a to b. Now, this is the minimum and this is its maximum. So, if we assume that a is less than b, trivial case, right? And let's just consider everything is positive everywhere, like here. Then obviously you can say that function f at x is greater than its minimum but smaller than its maximum, right? Which means that the whole thing would always be less than or equal to, if I will replace this with m, capital M, maximum, which results in times what? If this is m and it's out, I have only some of these, right? Now, some of these are some of these individual intervals, which is actually the length of the entire segment, right? This plus this plus this plus this gives you entire segment. Now, if you would like to introduce some, I would say, weird cases, like for instance a is not less than b, then you can't really put something like this. You need to put absolute value. But this is something which is not really important. What's important is that it's less than certain constant multiplied by b minus a in normal case when a is less than b, when it's a kind of a typical segment. Now, when it's really true, it's true whenever the maximum exists, right? But we have assumed from the very beginning that our function f at x is smooth function on a segment from a to b, which means at least it's continuous. And if it's continuous, it reaches its maximum and minimum on this segment, right? We are not talking about functions like, for instance, a tangent x, which goes to infinity at t over 2, right? We are not talking about this. Because then this integral, which includes this as a right boundary, we are not considering this. We are considering only functions which are smooth on the segment with both boundaries, so they reach its minimum and its maximum. And obviously the next property, which is completely similar to this one, is that it's greater than lowercase m, which is minimum, right? Because again, if this is greater, f at xi is greater than minimum, then the whole thing is greater than lowercase m multiplied by this. And we take it, factor it out from the sigma, and we have the lengths of the b minus a. So these are, again, two very simple properties of the definite integral. And now I will have something which is not exactly obvious, but very, very important. Well, it's kind of obvious. Look at this this way. If you, again, consider integral as the area under curve, and you divide it in two different pieces, like this is a, b and c. Now, area from a to c obviously is equal to area from a to b, plus area from b to c, right? It's an additive kind of a measure this area thing is, right? So we are assuming the following to be true. Now, how to prove it? Well, to prove it is really kind of simple. Let's consider a, b and c are in this particular order, right? a less than b, b is less than c. Now any partitioning from a to c obviously results in certain partitioning from a to b and then from b to c, right? So this particular sum can always be represented as sum from certain number from, let's say x0 to x, let's say 100 and then from x100 to x200, right? So the sum from 1 to 200 is equal to sum from 1 to 100 plus from 101 to 200. So that's basically the way how you approach this. So for any particular partitioning, you just represent this function, this sigma, as a sum of two sigmas, one sigma from 1 to, let's say, some number k and then from number k plus 1 to number n. One represents this, another represents that and that's why you're basically proving that sum of these is equal to, sum of these two integrals is equal to one total integral. Again, an area is very simple. On some other examples, like for instance, if you're covering the distance going into, in the car, obviously if you have the time period from a to c and you cover certain distance, time b is somewhere in between, then obviously your total distance from a to c is equal to distance from a to b plus distance from b to c. So it's kind of obvious on the intuitive level. That's probably a very important property which I will use when I will establish the connection between definite and indefinite integrals. So basically these are the most important properties of integral. Some of them absolutely trivial, some of them not exactly trivial but intuitively obvious like from this particular example. And by the way, you always can think about definite integral as area under curve. If you intuitively think about this, this way then you will obviously understand that all these properties are obvious. For instance, if you multiply function k by a factor of three, for instance. So it will be somewhere here. And obviously this area would be three times as much as this. So that's why you kind of decide that the multiplier can be taken out from the integral. Alright, that's it for today. I do suggest you to go to Unisor.com and take a look at the notes for these lectures. Other than that, that's it. Thanks and good luck.