Hi, welcome to the fifth week of the calculus course. Recently, we've covered the concept of derivative for both single variate and multivariate functions. Now it's time to turn over to the inverse operation for taking a derivative integration. First of all, one should always understand that if we introduce some operations, we are also should consider the possibility of introducing an inverse one and what it take us to actually define one. So what is an integration? Previously, we had a function and we thought about the instantaneous change with speed, but now we should think of an inverse task. We have instantaneous speed of the central function and we need to consider the function itself, find the function itself. In other words, we need to continuously sum all the instantaneous changes of our function to introduce the function itself, to reconstruct function. Well, that's what integration basically is, it's a continuous summation, but that's where it gets tricky. Let us just consider some basic example here. For example, let us use some well basic curve for applying this sine function that we have here. So the most basic theme for intervention is, for example, defining the length of this curve. The simplest one, well, if you just in kindergarten you were asked how long does this path takes, the only thing consumption like, "Okay, I'm going to put some dots here this one, this one, this one," and then I will draw a poly line. Well, in kindergarten, you do not know what is polyline but you know what the segment is, what a straight line is. So you draw it and then you calculate the sum lengths and you call it a result. Nowadays, when you are more aged, and well, you know much more, you can draw for example, much smaller segments or something like that. As a result you should understand that by introducing small and smaller segments, you will somehow be close to the actual length of the curve. But what is the natural answer here because ideally you should think about infinitesimal pieces here, infinitesimal segments. So you need to summarize all this in infinitesimal segments, and that is how you will get the curve. But, well, obviously in real life, it's not possible because you will always get an infinite number of these segments, but these infinite numbers also obviously are good for approximation. So why do you even bother and consider the continuous solution of integrals, if you can always go with something like closed thin sets. Well, among the thin set, there are always some non discrete variables. For example, time is essentially a nondescript variable. Usually always understand that in case of variables with a wide range, with multiple values since arranged, the summation is always tricky because you have a lots of [inaudible] in the sum, and it's in need of some nicer form as the sum, but you actually can get one out unless you consider this a continuous variable. That's basically the concept which is heavily used in probability theory and mathematical statistics which we will study further, when you basically manage a key change in transition from discrete variables, if you skipped random variables to continuous random variables. So that's what we're going to do. We're going to introduce ourselves to our basic toolbox for understanding the continuous summation of integrals. For the following week, we're going to do it in the following manner. We're going to start out with anti-derivatives and in other words indefinite integrals, then we are going to apply this concept to calculation there under curve, and substantially transfer this to the case of multivariate functions. Also, we are going to attach some numerical methods here, and you will be given a non-mandatory programming task by the end of the week. See you in the following video.