 Hi, I'm Zor. Welcome to New Zor education. Today, I would like to talk about some physical sense of anti-derivatives and integrals. Basically, this mathematical apparatus like differentiation and integration, they did not really appear in just vacuum. Somebody came up with these procedures. Well, the beginning of the process of developing these mathematical apparatus were people who were very much concerned about physical sense of whatever we're all doing. And primarily Sir Isaac Newton and Leibniz, they were in the beginning of mechanics, as we know it right now, classical mechanics. And all these procedures, differentiation and integration, were basically invented by them and some other people who developed it for the purpose of better explaining what's going on in the nature. So, being as it may, I would like to switch from the previous lecture where I was explaining basically purely mathematical apparatus of integration to something which is a little bit more physical, primarily to justify whatever mathematical apparatus we are using, whatever mathematical principles are in these purely physical concepts of mechanics, for instance. Now, you do remember that when I was trying to explain the principles of differentiation, I was talking about speed or rate of change as basically the physical principle on which differentiation actually is based. So, let me just go back and, well, not repeat, but remind you what exactly that was all about. Let's consider the R moving along certain trajectory. And our movement is basically controlled by certain mathematical function which tells us how much distance we have covered from the moment t equals to zero to current moment t. So, let's say we started here. So, for every moment t, this is t equals to zero. So, for every moment t, we know how much distance we have covered during our trip. So, and that's what actually s of t is the length of this piece. So, this is moment t. Now, what we did next, we were talking about the speed. And first, we decided that if you have certain interval of time from t to t plus delta t, then during this time, we have covered the distance from s of t to s of t plus delta t. So, we have covered the difference in distance, is basically the length of this little thing. So, s of t plus delta t minus s of t is equal to the distance we have covered. Now, the t plus delta t minus t is the time during which we have covered. So, this basically gives us some kind of average speed during this time from t to t plus delta t. And if my delta t goes to zero, the limit of this, so this particular interval of time becomes smaller and smaller, the limit of this is an instantaneous speed at moment t. So, let me just put the word limit. So, that was our justification for introducing a derivative because this limit is actually a derivative of function s of t by t. Sometimes we do it this way, sometimes we do it this way. Different notation, but anyway. Now, this notation is a little bit more common place because it actually reflects what exactly we did. So, this is a differential, which is a limit of this infinitesimal variable divided by this infinitesimal variable. So, the d basically basically encapsulates in itself this fact of limit. So, these are concrete values and these are infinitesimal variables and we have a ratio. Now, let's recall another very, very long time ago, I had a lecture about harmony. Mathematicians do strive to have harmonious constructions. And one of the, well, obviously it's subjective, but I do think it's one of the very interesting properties of harmony is if you have moved from A to B, you have to be able to move from B back to A. So, the operation which does something, and in this case we have an operation of differentiation from the function s of t, we got the function derivative of s of t, right? So, this is operation from one function we derive another and obviously if we can move back from the derivative back to original function, that would make actually the whole construction more harmonious. And we did decide that this is an operation. It's called integration. I defined it in the previous lecture and that actually contributes to this harmonious thing. But now let's go back to the physical entity of this. Now, what happens in this case? If our average speed during this interval delta t is equal to, by the way, this thing obviously is delta s of t, right? It's an increment of the function. It's equal to increment of the function divided by increment of the time. Then increment of the function is equal to this average speed during this particular period of time delta t times delta t, right? From this, I derive this. Now, again, we're talking about physical sense of whatever we're doing. So far so good that this is an average speed during this time interval. Now, how can I construct the function s of t? s of t is the total time I have covered from moment zero to moment t, right? This one. And it contains many small intervals. So if I will divide this into small intervals, and on each interval I basically can do this, and then I will summarize all these small intervals, that would be my total distance, which I have covered, right? So basically, a sum of these would give me the total distance. Now, obviously, this is not precise because this is an average speed. This is an average speed during this interval. But if the time interval becomes smaller and smaller, and smaller, so the number of these little intervals is increasing, my precision of the result of summation of these things would be better and better approximating my real distance, which I have covered, right? So basically, we are talking about summation of something. If my delta t is converging to zero, so each one of these is infinitesimal, but the number of these will be growing into infinity, then my approximation to s of t would be better and better. So what did those people who invented the integration actually what did they do? Well, and primarily it's Newton and Leibniz as I was saying before. Well, they actually used this as a prototype for notation of the integration. So what they did is instead of this sigma, they imply that this is a sum of infinite number of terms here. So they used the letter s, but they stretched it very much and called it an integral. Now this is, obviously, as delta t goes to zero, this would be my instantaneous speed, which is actually a derivative, right? This is a derivative. And this becomes, instead of delta, they use g to indicate that this is infinitesimal. Basically, very similar to whatever we did here, right? So let me just replace this with this. And here we go. And this is now my exact distance. Well, obviously, later on they realized that to make this actually precise, you do have to add constant. We were talking about this before, but in any case, well, rather here, it doesn't really matter where I add it here. Now, what's the meaning of this constant? Well, basically, I would interpret it as the distance covered before my moment t is equal to zero from some other point. And obviously, this piece remains exactly the same. So in any case, I wanted to justify that this is not just, you know, somebody came up with this particular notation, as I was basically suggesting that let's just forget about the meaning of this, just accept it as a notation of integration. But actually, it does have this physical sense underneath. There is a basis for this. And the basis is this constitutes an instantaneous speed times infinitesimal interval of time. And if you will summarize the infinite number of infinitesimal variables that would actually constitute the original function distance as a function of time, which we started from. So consider integration as basically an operation which is delivering my distance based on speed. And obviously, it can be expanded to many other processes where it's not really like a speed, but something like a rate of change of something. And the result, so this constitutes the rate of change of something. And this is a result of that change during certain amount of time. So that was in the beginning of differentiation and integration as Newton and Leibniz primarily these two, and then our best and many other mathematicians. That's exactly what they put as a physical foundation of all these mathematical apparatus. Now, we cannot really say what was the physical or natural foundation for basic operations like addition or multiplication. We were just thinking about maybe people wanted to count something. But in case of integration and differentiation, we do have the physical need and real people who felt that this is the apparatus which they need to analyze mechanics of movement and any other rates of different processes which are happening in nature. So this is something which happened during our known history. It's documented. So there are real people, there is a real need for this and the real mathematical response to physical mechanical whatever and other natural needs. All right, that's basically all I wanted to say about integration. I wanted to put it on a certain more natural, more more physical foundation. So you do not really think about integration and differentiation as just two abstract operations. No, these are very much used in like everywhere in whatever we are doing, starting from just opening the faucet in the in the bathroom and ending with maybe spaceships everywhere. All these mathematical operations, integration, differentiation, etc., they're all used very much in our day-to-day life. That's it. Thank you very much and good luck.