 Hi, I'm Zor. Welcome to a new Zor education. Today I would like to talk about a few different applications of integrals, primarily of some kind of physical kind applications, which is not a surprise actually because the whole concept of integrals was developed by people who were quite familiar with physics like Newton and Leibniz. Well actually they have created physics as we basically know it and learned in school. So it was created from real practical needs. Now the previous lecture was dedicated to area under curve, which seems to be a little bit more mathematical than practical. Today I will talk about certain practical problems which lead to the concept of definitive integral. Well this lecture is part of the course of advanced mathematics for teenagers presented on Unizor.com. I suggested to go to this website actually to listen to this lecture because the site has not only the references to video lectures but also very detailed notes for each lecture and exams for those people who are willing to take exams. Not for every lecture but for many of them. Okay so we will talk about three different problems and in all three of them we will show how the same concepts which we were talking about when dealing with the area under curve actually are applicable to these cases. Okay the first problem is how to calculate the distance covered by let's say a car. Okay so if the car drives with a constant speed from the moment time is equal to a to time is equal to b and we know that speed let's say the speed is v whatever it is meters per second and these are times in seconds obviously. We know how to calculate the distance covered we just multiply the speed by the period of time during which this car was moving so that's easy. What if our speed is not constant but instead a function of time and let's say we do know this function of time and again our problem is exactly the same. We have to calculate how much we have covered from the moment of time t equals to a to time t equals to b if at any particular moment we know the speed of the car and obviously we assume that this speed is smooth so we don't really have you know during certain period of time the speed is one and then in immediately next moment of time the speed grows to something somewhere so it's smooth function continuous differentiable whatever you want whatever we need actually. Okay so how can we approach this problem? Well it's not easy and again Newton, Sir Isaac Newton and Leibniz as well were basically facing the same problem and what they have decided to do is very similar to what we were talking about when presenting the area under curve. Let's in this particular case divide our interval into smaller parts from a to b we have divided into small intervals of time. Now since our function is relatively smooth if the interval between let's say t i and t i minus 1 if from t i minus 1 to t i if this interval is relatively small then the speed on the left margin of this interval or on right margin or the maximum speed during this time or a minimum speed of during this time they are very very close so let's assume that to approximate the time the distance we have covered from time t minus t i minus 1 to t i which we will actually call delta t i so during this particular interval of time we assume that since the speed is not really changing very much considering this interval is small we can say that approximately the speed during this period of time is this since we know the function v at any moment of time we know it at this particular moment of time which is the right margin of this particular interval right well we can as well take the left boundary of the interval and take the speed and consider it to be constant so considering this is a constant speed during this interval we will have certain approximation to the real distance covered during this particular time so let's call it delta s i which is speed at moment t i multiplied by the length of the time so this is not exact distance which we have covered during this interval of time but as an approximation it's good enough now the approximation it's intuitively obvious will be better if this interval is narrower so the speed will not change too much right so let's consider that as we summarize these distances by i and we will take the limit of this as number of intervals goes to infinity and the maximum of delta x goes to zero so even the widest time interval still shrinks to zero so considering this is our process it's reasonably to assume that this particular sum which constitutes an approximation of the distances covered at each particular interval of time and summarized together will be a good approximation and at the limit it will be the exact distance which will be covered by this particular car so we cannot really escape but just defining the distance which is covered by the car which moves with a variable speed through some process like this and again it's very much similar to our approach when we were calculating the area under the curve what is area under the curve the definition actually we have come up with is let's divide it into different pieces and let's increase the number of these intervals while the width of the widest interval converges to zero and summarize these pieces and that will be the area so in this case we will do exactly the same thing except this is a very physical part and we can actually measure our car at time t is equal to a is at one particular point and then let's say we move along the straight line with a variable speed and at the moment t is equal to b we are at some other point and we actually measure the distance and this distance should be exactly the same as this one so basically it's a combination of something which we define and something which we can verify but in any case you should understand that this is a reasonable approach if we do not have a ruler which helps us to measure the distance from the beginning to the end but we only have the speed at each moment of time then we use this approach since we cannot measure physically the distance we have only the speed then we have to really do something like this alright so the formula which we have come up with is exactly similar to the formula which we were using at calculating of the area under curve and it's important because other two problems which I am going to present will also lead to the same kind of formula and that would actually show how important this particular expression actually is ok so next problem next problem is about draining the tab ok let's say this is your tab and this is your drain now as soon as I open the drain the water starts going down right now the speed let's say liters per second of the flow of the water going down the drain is changing because if the water is high the pressure is high and the speed would be probably greater the water is lower the pressure will be lower so the speed of the water will be also lower by the way if your opening is not here but let's say here on the side you will see that it will go with a definitely higher speed out of this and you can actually measure it out of this opening so in any case we have something which we can call a speed of the flow of the water measured in let's say liters per second and we do know this particular function and it's a smooth function and all we need to know is how much water would flow out from the tab from the moment a to b how can we measure it we will do exactly the same I mean it looks like it's a different problem but we will use exactly the same approach we will divide this particular period in different certain number of intervals the smaller the better obviously right now we assume that during the period from from t i minus first to t i speed is more or less constant or I mean it does change but not significantly so as an approximation we can use the speed let's say at the moment t y t i as the speed which the water was flowing out during this relatively small period of time and the amount of water would be this right where delta t i is t i minus t i minus 1 and this is speed at the moment t i what do we do next well we summarize it and that gives us a total amount of water which has been flowing out but not exactly it's just an approximation to go into exact amount of water we obviously have to go to a limit of this sum as n goes to infinity and maximum of delta t i converges to zero so we are dividing with a finer and finer intervals smaller and smaller shorter and shorter intervals of time and that's why our approximation becomes better and better and better as in the previous case with the distance we can obviously measure what's my level of water at moment t is equal to a what's my level of water at t equals to b and calculate exactly the volume which has been flowing out and we do understand that it should actually be equal to this one and again this is exactly the same kind of expression which we are interested in if we don't know really how to measure this particular volume consider the following what if you have not just a straight tap something like very strange shape then even if you have the level at t equals to a and then level of t equals to b whatever here is this volume you can't really calculate it easily right so you still need some kind of a mechanism the speed which you can actually measure then you will be able to calculate this volume using this approach because you can't even measure it by the way in the previous problem if it's a straight distance that's fine you can actually measure where the car was in the beginning at the end if the way the car moves is really very kind of curvy that's not so easy because you really have to know how to measure the curves the lengths of the curves which is a completely different story and we did not really talk about this yet that's the difficult thing to do to define it but this approach will always give you the result so you have to be able to calculate the same type of expression which we have obtained from this particular problem and let me give you one more problem which gives exactly the same kind of a formula that would be the final how many different examples do you need to convince you that this is really a very important kind of approach in formulas okay, we are talking about volume of solids of revolution I'm not talking about revolution in a political sense I'm talking about the following let's say you have some kind of a curve and defined from A to B and this is the curve which define a three-dimensional body obtained if we revolve this around the x-axis so what will be if you will spin this particular thing around the x-axis it will be certain three-dimensional solid and we need to determine its volume let me give a simple example if you have a straight line as f at x from A to B and you are revolving this around the x-axis what will you have? well, you will have a cylinder, right? now this would be the radius of this cylinder and this would be the height of this cylinder, right? so the volume would be able to pi r square h, right? pi r square would be the area of a circle which is the right base of this cylinder the same as the left base of this cylinder and instead of h I actually should put B-A and instead of r I should put function square of let's say B or A doesn't really matter because they are the same, it's a straight line parallel to x, right? so that would be my volume now if you have a slightly different case again relatively simple, a little bit more complex if you have a straight line between A and B what will be if you start revolving around the x-axis well you will have a truncated cone, right? if it goes up to this, from this to this and you spin it around the x-axis you will have a cone, but if it's from A to B you will have a truncated cone, right? without the top that's what you will have and again you know how to calculate the volume of this particular case what I'm talking about in this case a more general case is I have no idea what this function is and I do know that it revolves the whole curve revolves around x-axis and we need to know the volume how can I calculate this? well the random formulas, right? so I will approach it in exactly the same way so from A to B I divide it in different small intervals which I will use x instead of t it's not a time, it's distance so x i minus 1 and x i so A is x 0 then x 1, x 2, x 3 and B is x n now if I will consider only this piece from x i minus 1 to x i there will be a thin kind of a slice, right? something like this, right? now this is not a straight line this is this which is kind of a curvy right? whenever it spins around so this is our x-axis, right? so this is the piece I turn it 90 degrees and that would be the result now this is not a cylinder although it's very much close to a cylinder the smaller my interval between x i minus 1 and x i is the thinner is my slice and that's why these little boundaries I can just basically ignore them in approximation and say that the radius of this cylinder is let's say function of x i, right? this one because there is no big difference because this is f of x i and on the bottom you have f of x i minus 1, right? but they are very close to each other because we consider the function f of x is smooth enough so the smaller height of this slice and height is what? the distance between x i minus 1 and x i, right? so the height of the cylinder is x i minus x i minus x i minus 1 which is delta x i and the radius of this cylinder is f of x i, right? so the volume of this cylinder is pi r square h, so it's pi f square of x i times delta x i now the total volume of all these slices is approximately a sum of these and if I would like to really know the volume of the solid which is a result of a revolution of this particular graph, this curve I have to take the limit and the limit again is as number of slices number of intervals goes to infinity with the largest among them converges by lengths to zero so again I have exactly the same formula some kind of a function at value times the interval of argument and then the limit so more and more important becomes this particular construction well in this particular case function is actually not f of x it's pi f square of x but doesn't matter what kind of a function it's a function we have basically a very general very general construction which looks like this limit sum of function of x i delta x i i from 1 to n n goes to infinity and maximum of delta x i goes to zero so this is a construction which we are dealing again and again and again and this is exactly the one which we are dealing with area of curve area under the curve and in that lecture when I was talking about area under curve I was talking about two extremely important considerations about this particular thing number one if function is relatively smooth it's continuity maybe differentiability etc I mean that's something like exactly completely I said it's sufficiently smooth to basically justify whatever the logic we were using so for smooth function this particular limit exists regardless of the way how exactly I'm partitioning my segment from A to B into individual intervals so as long as number of intervals goes to infinity and the maximum of the interval is converging to zero the limit would exist and it will be exactly the same so there is an existence and there is a uniqueness this is a number basically it's a certain number which is number one it exists number two it's unique regardless of the way how I divide my interval into pieces will lead us to the next lecture which will be a definition of the integral definite integral and basically it will be I can tell you right now this limit is the definite integral but that will be the next lecture and I will basically talk about properties of this integral etc well for now that's it thank you very much I do suggest you to read all the notes for this lecture on Unisor.com and well basically I always encourage you to take the whole course from the beginning but this is kind of the end of this course already alright so that's it good luck, thanks a lot see you next time