Understanding Riemann Integration Concepts

Hello everyone, so welcome to my playlist on single variable calculus. So today we are going to see the concept of Riemann integration. So we will see how Riemann gave the idea that how to approximate the area under the curve using rectangles. And once you see that you will you will realize that derivative and integration, they are not opposite of each other. Because usually when I ask students, what do you mean by integration, they say it's an antiderivative. that is opposite of derivative no which is not the case in some good good cases it happens and that's where fundamental theorem of calculus comes into picture so that we will see once in my coming lectures but today we will see the concept of integration and it will help you to realize that derivative and integration they are opposite i mean they are not opposite of each other they are totally different so i will explain the concept after that we will take the help of geogebra software because usually softwares are good to visualize the things. And after that, we will again come back to blackboard and I will tell you how to write down the things because I have seen usually the idea is clear. But when I tell students to write it down, they make silly mistakes or they get confused what to write how to write. So this lecture is divided into these three phases. So let's go one by one. Let's try to see what is the notion of or what was the idea of Riemann. Okay, so suppose you have a function from close a b to r, then question is what is integration a to b f of x dx, we will see this why this symbol also comes. I mean how this symbol came into picture, we will see that later. But then what is this everyone knows that this is a number because you have seen in your high school, you have solved integration. So you know that it's a number and you know that it's a under the curve. But then how it is a under the curve? That's what the question is. Okay, so that's what we're going to see. So this is my x axis. This is my interval a b. And suppose this is graph of some function, I am taking the simplest one. So as you can understand this properly, okay. So what Riemann said, the first thing you have to do is you have to partition your domain. So partition means what? So let me call P as the partition. So let me divide into partition means what you divide your interval into sub intervals. Okay, so let me call this as x naught. Let me call this as say x1. Let me call this as x2 and let me call this as x3. So I'm dividing into three sub intervals as if now how to do it in n sub intervals that I will tell you later on but for understanding purpose let's play with the basics. Okay now so what is my p it is x0 which is my left end point then x1 x2 and x3 which is my right end point. So this is p is the partition of my interval a b. Okay and this xi's you will see this word many times this xi's are called as the nodes of the partition. Okay x naught x1 x2 x3 they are called as the nodes. Okay now what you do is you consider first this sub interval. So let me take a different chalk. You consider first this sub interval. So this is till here you have x1. and this is what you have then again the same thing you consider the next sub interval this is what you have and the last sub interval this is what you have Okay, the what Riemann said you in this first sub interval, you look for the point where function is taking the minimum value, okay, you try to look for the point where function is taking minimum value. So this is the graph. And you can see that at a or at x naught function is taking the minimum value f of x naught So when you multiply this height into this length, how much is this length? So this length is x1, this length is x0. So how much is this length? x1 minus x0, right? So the length is x1 minus x0. into the height. So length into length into breadth will give me area of this sub rectangle. So what is this area it is nothing but let me write it over here. So I am doing f of x naught into x1 minus x naught. Okay, cool. Let's go for the next sub interval, you look for the point where function is taking minimum value. So let me call this as say, c1. And at here you are having the minimum value, right? So this is my height. And this is my length. So length into breadth will give me area of this sub rectangle. So what I'm doing is I'm doing f of c1 into x2 minus x1, the length of this sub interval. Similarly, go for the last sub interval at x2, you can see function is taking the minimum value. So this is f of x2. So f of x2 into this length will give me area of this sub rectangle. So what I'm doing is f of x2 into x3 minus x2. So I concentrated in the sub intervals and I found the area of the rectangles and when I sum it up will I get the actual area under the curve answer is no because as you can see this area got missed this this and this area also got missed. So, this is not giving me the actual area under the cover. But then he gave a name to this number and what name did Riemann gave? He called this as LPF or some author also write as LFP. L means what? The lower sum. Why lower sum? Because we are looking for the point where function is taking the minimum value. So, this is called as the lower sum of a function for the given partition P. Okay or lower sum. sum for a partition P for given function f. Usually I prefer this notation because first you should know function and then you partition right so yeah it's depend doesn't matter you can use any notation. So lower sum for given function function for a given partition P and this is called as my lower sum. So whenever someone talks about lower sum, that means you have to understand that over the sub interval he or she is looking for the point where function is taking the minimum value. So okay lower sum is not giving me the actual area under the curve. So, let's go for the other side other side as in let's try to find the upper sum. That means let's try to look at the points where function is taking maximum value and let's try to see whether that will give me the area under the curve. What do you think? Let's check it out. So now let's go for the upper sum. So now in this first sub interval you look for the point where function is taking the maximum value. So as you can see at here the function is taking the maximum value. So let me call this as a c1. So if I do f of c1 into the length of the sub interval x1 minus x0. So what I will get this is the maximum value. So this is my length. So and this is my height. So, what will I get? I get area of this sub rectangle. Do the same thing for the next sub interval for x1 to x2 at which point function is taking maximum value. You can see at x1 it takes the maximum value. So, therefore, So like this is the area of the sub rectangle the length into the height and for x2 to x3 at x3 takes the maximum value so if I do this so this is the height and this is the length so I get area of this sub rectangle. So what I'm doing is plus in the next sub rectangle it was at x1. So f of x1 into x2 minus x1 plus here it was at x3. So f of x3 into x3 minus x2. So this is the area of the sub rectangles and we are summing it up. And since we are taking the maximum value of the function this sum is called as u. fp upper sum for a given function over a partition p. So is ufp the actual area under the curve? So lfp was not giving me the area under the curve. The area got missed. What about ufp? Is it also giving you the area under the curve? No. Why it is not giving you area under the curve? You can see you are getting extra area. You are getting the extra area. So even ufp is not helping me. Ok, so neither LFP nor UFP is helping me. Okay, and what is the connection between them you can see your LFP is obviously less equal UFP because L is the lower sum so there the area got missed and U is the upper sum you are getting extra area. So obviously this area is bigger than the this area. So LFP is less equal UFP. Okay, so that was one connection we saw between the lower sum and the upper sum. And we also saw that neither lower sum nor upper sum is giving me the actual area under the curve. So what to do? Then comes the refinement into the picture. So let's try to see what is the concept of refinement and how does it affect these areas. Okay, so we saw that neither lower sum nor upper sum was giving you or is giving you the actual area under the curve. So now the question is what to do? So Riemann said okay you refine this partition. Refine as in what you insert some more nodes into this partition. So let me call my new partition as q. So what I'm doing so x naught. So let me rename this for simplicity. So let me call this as x1 x2. Let me call somewhere here x3 x4 or some or let me call this as x5. Okay, so my x naught is here. Then x1, x2, x3, x4 and x5 is my right hand point. So what is my Q? Q is also the partition of the interval a, b. And what can you say about p and q? One can observe that p is a subset of q because q contains p. And since q has more nodes, that means what Q is doing, it is making intervals much more smaller. that means called as a refinement. So therefore, our Q is called as refinement of P refinement of partition P. Okay, so this is what your Q is. Okay, now let's try to find upper and lower sum. Okay, so let me take a different chalk. Okay. Now do the same thing. You concentrate on the interval. Okay, so make sure the color is different. Okay, so the purple one was the area for the partition. and P now we are going to find the lower sum but for Q partition and let's try to see what is the connection okay now this is my first sub interval x0 to x1 at which point function takes minimum value at x0 function is taking the minimum value so this is my breadth and this is my length so this is the area I get okay so it is let me write down with the white chalk so it is f of x0 into x1 minus x0 Let's go to the next sub interval x1 to x2 at which point function is taking minimum value at x2 it takes a minimum value right this much is the curve. So at x2 function is taking the minimum value. So yeah, this is my height and the length so I get this area. So what I have is f of x2 into x2 minus x1. Now you compare earlier when we found the lower sum for P partition we got this much as the area then we did the refinement and now if you find the lower sum we are getting If we are getting this much area and this much extra area, which is a good thing, right? So with Q I'm getting much more extra lower sum. I mean, I'm coming towards the closure of the area under the curve. Okay, so let's do the same thing over here. If you concentrate on the interval x2 to x3. So again let me call this as say d1 at d1 it is taking minimum value so this is my height into breadth so this is the area of the rectangle then x3 to x4 at x3 function takes minimum value so this is the height this is the length so I will get this as the area of the sub rectangle here also if you observe initially I only got this much as the area okay initially I only got this much as the area But after refinement, I'm getting much more extra area. And the same thing works here. Okay, so here, I have not inserted any node. So yeah, here it remains the same. If you want to insert not an issue. Let me call this as x five and x six. So x five, and let me call x six as my b. So here if this is my node so this is what I have so over here this will be my lower sum because at x4 function is taking minimum value and here at x5 function takes the minimum value so this is the area I have so what you can observe so when I sum it up plus this area plus this area plus this area plus this area and when I add it up f of x0 into x1 minus x0 f of x2 into x2 minus x1 here f of d1 into x3 minus x2 here f of x3 into x4 minus x3 here f of x4 into x5 minus x4 and here f of x5 into x6 minus x5 and when you add it up this is my lower sum for the function for the partition Q. So, this is the lower sum for the partition Q and you can see that this Q is giving me the area which I am getting through Q partition is more than the area which I got for the P partition. So, the relation which you can get between P and Q this is the first relation and another relation is your lower sum. For the function for the given partition P is less equal the lower sum for the function for the partition Q where Q is what it is the refinement of P and as you can see with Q and getting the extra area. So there's a good thing right so can keep this one in equality in mind, but again the same thing this is not giving me the actual area under the curve. So what should you do now? So, the idea is you further subdivide this interval into further smaller sub intervals. That means you increase the number of nodes. So first you had three then you had six then you make suppose hundred nodes. What do you think your lower sum will keep on increasing. Your lower sum will keep on increasing. What about the upper sum? So let's try to see the behavior of upper sum for the partition Q. Okay so we saw that after refinement Even Lpq sorry Lfq was not giving me the actual area under the curve. Let's try to see the behavior of Ufq means upper sum for the function over the partition q. Now here also you do the same thing. So let me take a different chalk. Okay so it's a blue color. Over the first sub interval look for the point where function is taking the maximum value. So here you can see at x1 is taking the maximum value. So this is the height. Or okay let me. do this some nice partition so let me call I think my x1 was here if I am not wrong. Okay so you do the same thing now for this interval you look for the point where function is taking the maximum value so here you can see let me call this as d1 at d1 it is taking the maximum value so the maximum value into the height so I get the area of this sub rectangle. So what I'm doing is f of d1 into the length x1 minus x naught. Cool, go for the next sub interval x1 to x2. So at this x1 I'm getting the maximum value. So this is the height into the length. So what I'm doing is I'm doing f of x1 into x2 minus x1 and now you see the magic the area got decreased after the refinement obviously u fp was not giving me the actual area under the curve I was getting larger area but after refinement upper sum got decreased okay let's try to do the see the same thing in the next sub interval for x2 to x3 at x2 it takes a maximum value so this is the area of the sub rectangle length into the height into the length and then from here x3 to x4 okay This is the area of the rectangle I get. Here also if you observe little bit of area got missed. Okay same thing x4 to x5 here is the maximum value. So maximum into the length will give me area of this sub rectangle and from x5 to x6 maximum into the length will give me area of this sub rectangle and here also you can see this area got missed. So when I add everything plus this area this area this area and this area this i called as u f cube upper sum of a given function for over the partition cube okay so what good thing is happening and let me write down one more inequality over here i don't need this so what we know that your you your LFP is less equal LFQ LFQ is obviously less equal UFQ and UFQ is less equal UFP Right, this is what we have. So, you had a curve over here, you wanted to find the area under the curve. What you do? You do the partition. For the partition you find the lower sum area of the sub rectangles, you find the upper sum area of the sub rectangles. Those were not the area under the curve. So, you do the refinement. Lower sum increases. So, this is the curve over here. So, upper sum, sorry, lower sum increases and upper sum decreases. That we are still not giving you the actual area under the curve. So you again do the refinement. Then again the lower sum will increase upper sum will decrease. You again do the refinement. You keep on doing this. So when you keep on doing the refinement you increase the number of nodes. Then your lower sum will keep on increasing. Your upper sum will keep on decreasing. And when they match. then that is actually the area under the curve. Okay, so that's what the idea Riemann gave. So when first you had six three nodes, then six nodes, then you put 100 nodes, then you put 1000 nodes, then you put 1 million nodes, you put one crore nodes, you put 1 million zillion of nodes, so keep on adding the nodes. That means you make the length of the sub interval smaller and smaller, you make the length of the sub intervals smaller and smaller. And actually that there is one more node for this one more notion called is the mesh of P mesh means what the I mean the distance between the nodes become smaller and smaller. So P is the partition mesh of P means what the distance between the nodes should get smaller and smaller. So keep on inserting the nodes. Keep on inserting the nodes. that means the length of the sub intervals getting smaller and smaller and when that happens that means your n is going towards infinity that means so when your lower sum limiting case of the lower sum is equal to the limiting case of the upper sum and when they are equal we say that the function is remain integrable okay so what i'm trying to say over here is so what you had you had l fp less equal L F Q suppose R is the refinement of Q this is less equal L F R and you keep on doing that this will be less equal U F R. less equal u fq less equal u fp that means what as you keep on doing the refinement your lower sum will keep on increasing your upper sum will keep on decreasing and when they matches we say that the function is integrable and whatever that matching value is that is nothing but integration a to b f of x dx okay so let's try to understand this using geogebra software so when you go to this geogebra software at the i mean here you can see this search classroom resources so if you simply type remand sums i mean many good people have just made their activity public so that everyone can see so you can take whichever you like okay so i mean there are plenty of them so let's take let's go for this one suppose so now here you can see here is a function given to you here they have taken n 10 nodes n is equal to 10 means 10 nodes you can see there are 10 sub intervals you can play with this function and now here you can see as i increase the node see the pink one is the lower sum and the blue dark the blue one is the upper sum so you can see that upper sum is giving you more area lower sum is giving you lesser area okay this gray part missed away when you increase the nodes you can see the upper sum gets decreases and the lower sum increases and that means this rectangles are coming closer and closer to the curve so you can see if i try to zoom but still there is a gap okay so you can see there is a gap so you again increase the partition you keep on increasing like here there is only till hundred but yeah as you can see there is still gap so to further increase the node so you make 1000 then again there will be a gap. So, you keep on increasing the nodes your lower sum will come closer and closer to this curve and the upper sum will again come closer and closer to the curve and that's what I tried to tell you earlier. So, you keep on increasing the nodes your upper sum will come closer and closer to the lower sum. Now, you can see that the rectangles are very close to the curve. They are not equal to because when you zoom in or zoom out whatever. So, you can see that there is a gap. okay so for smaller nodes you can see gap properly with your naked eye when you increase the nodes you need to zoom to see that actually yes there is a gap so you further increase the nodes and like that when so keep on increasing that means a norm goes to zero or your n goes to infinity if the lower sum is limiting case of the lower sum is equal to the limiting case of the upper sum you say that your function is an integrable function. here the good thing is you can click here and you can play with any function which you want we played with x square so you enter x square this is you get the graph of x square if you want sin x you get the graph of sin x you can play with sin x as well so you keep on changing the function and you can play with it so that's the good thing about this okay so now let's see how to write down the things properly okay so last part of the lecture how to write it down okay so you have a function from close a b to r the first thing is you first partition your domain so let p equal to x naught the left hand point x1 x2 up to xn the right hand point let p be a partition of the interval a b and your x i's are called as the nodes of the partition p not writing that statement but yeah you should write it down okay so once you have the partition what is the next thing you do you find the lower sum so how you find lfp you It is the summation right? You are summing up the areas. So you are multiplying length of the sub interval with minimum value of the function where i is going from one to n because over each sub interval you are finding the area and you are summing it up. And what was your f of ci? Where f of ci is equal to what you are doing you are having a sub interval xi minus 1 comma xi over that sub interval you're looking the values of x and you're taking the functional values so f of x where x over is x is over this sub interval and you're taking the minimum of that right so over that sub interval you look for the point where function is taking the minimum value so you're taking the minimum of the functional values over that sub interval so that is what my f of ci is so that how one should write okay Then comes the UFP. What is UFP? It is summation i from 1 to n f of z di into xi minus xi minus 1. And now here where my f of di is what? It will be the maximum of the function f of x where x is in this sub interval. So that is what my LFP and UFP is. So you had the lower rectangles, you submit up that is my LFP, you have the bigger rectangles, you submit up that is my UFP. But as we saw, they were not giving us the area under the curve. So we did the refinement that means what we inserted more and more nodes. And when we did that, lower sum increased, upper sum decreased, you further keep on doing that you can keep on increasing the nodes, lower sum will keep on increasing, upper sum will keep on decreasing and they should match. Okay, that means what? you are taking n going to infinity you are keeping on doing this process. So, so when you keep on doing some process, then the limit comes into the picture. And therefore, now let me call my lf let me call lf as limit n going to infinity of lfp that means you take n going to infinity. But I would prefer to write norm of P or the mesh of P goes to zero. That means what like the initial you had a B interval whose length is B minus a then you inserted nodes so the length became smaller. You further insert the node so again the lengths become smaller. If again enter the nodes again the length becomes smaller that means the length should go to zero. Okay, I prefer this notation not this. Why so I will tell you maybe sometime in the next lecture but yeah anyways at least for engineering background I will say this is sufficient even if you write this okay but if you are doing pure mathematics then make sure this point is clear okay. Now what is my uf it is again same thing limit n going to infinity or you should write mesh of p going to 0 u f p and what I said at the end they should match that means if your LF is equal to UF we say the function is integrable that means we are getting the actual area under the curve okay so let me call this as number whatever the number they are same so this number is nothing but this value integration a to b f of x dx So that's how you find the integration. Okay, so at the start, I told you that integration and derivative are different. And now you can see derivative is a slope of a tangent line, and integration is an infinite sum, right? Because this was a finite sum. And then you're taking n going to infinity. So this is what is integration, it is an infinite sum, you're adding infinitely many numbers, and it is giving you a value. So integration is an infinite sum derivative is a slope of a tangent line. So you can see they're totally different. Okay, that's one thing. And that's how this also comes up like this integration is nothing but actually s only. So I mean, the s is nothing but the sum. So integration is nothing but the sum. And they just elongated this s. So when you elongate the s, you get the symbol of integration. So that's how this this represent the sum. Okay, this dx means what you're doing it into the small small, small, so x you are doing it dividing into the small small interval that means the change in x. So therefore this is the functional value f of ci f of di and this was the length of the sub interval and you are taking the infinite sum. So this is how this notion or this notation came into the world or came into the picture. So yeah so that's all about the idea of Riemann. Now one can have many questions. I mean how to solve this by taking a proper example. So in the next lecture, I will take a proper example. We will solve that integration using this method, this algorithm, then we will also see example that which are not integrable, okay, not all functions are integrable, there are functions which are not Riemann integrable. So we will see that. And we will also see some examples on Riemann sum, lower sum and upper sum. Okay, so yeah, so when this number exists, we say the function is Riemann integrable, or the integration of f exist, this thing. Okay, so that's all about the idea of Riemann. If you have any doubt, you can ask me in the comment section and if everything is clear, do not forget to like, share and subscribe. Thank you.

that is opposite of derivative no which is not the case in some good good cases it happens and that's where fundamental theorem of calculus comes into picture so that we will see once in my coming lectures but today we will see the concept of integration and it will help you to realize that derivative and integration they are opposite i mean they are not opposite of each other they are totally different so i will explain the concept after that we will take the help of geogebra software because usually softwares are good to visualize the things. And after that, we will again come back to blackboard and I will tell you how to write down the things because I have seen usually the idea is clear. But when I tell students to write it down, they make silly mistakes or they get confused what to write how to write. So this lecture is divided into these three phases.

So let's go one by one. Let's try to see what is the notion of or what was the idea of Riemann. Okay, so suppose you have a function from close a b to r, then question is what is integration a to b f of x dx, we will see this why this symbol also comes. I mean how this symbol came into picture, we will see that later. But then what is this everyone knows that this is a number because you have seen in your high school, you have solved integration.

So you know that it's a number and you know that it's a under the curve. But then how it is a under the curve? That's what the question is.

Okay, so that's what we're going to see. So this is my x axis. This is my interval a b. And suppose this is graph of some function, I am taking the simplest one. So as you can understand this properly, okay.

So what Riemann said, the first thing you have to do is you have to partition your domain. So partition means what? So let me call P as the partition.

So let me divide into partition means what you divide your interval into sub intervals. Okay, so let me call this as x naught. Let me call this as say x1.

Let me call this as x2 and let me call this as x3. So I'm dividing into three sub intervals as if now how to do it in n sub intervals that I will tell you later on but for understanding purpose let's play with the basics. Okay now so what is my p it is x0 which is my left end point then x1 x2 and x3 which is my right end point. So this is p is the partition of my interval a b. Okay and this xi's you will see this word many times this xi's are called as the nodes of the partition.

Okay x naught x1 x2 x3 they are called as the nodes. Okay now what you do is you consider first this sub interval. So let me take a different chalk. You consider first this sub interval. So this is till here you have x1.

and this is what you have then again the same thing you consider the next sub interval this is what you have and the last sub interval this is what you have Okay, the what Riemann said you in this first sub interval, you look for the point where function is taking the minimum value, okay, you try to look for the point where function is taking minimum value. So this is the graph. And you can see that at a or at x naught function is taking the minimum value f of x naught So when you multiply this height into this length, how much is this length?

So this length is x1, this length is x0. So how much is this length? x1 minus x0, right? So the length is x1 minus x0. into the height.

So length into length into breadth will give me area of this sub rectangle. So what is this area it is nothing but let me write it over here. So I am doing f of x naught into x1 minus x naught.

Okay, cool. Let's go for the next sub interval, you look for the point where function is taking minimum value. So let me call this as say, c1. And at here you are having the minimum value, right? So this is my height.

And this is my length. So length into breadth will give me area of this sub rectangle. So what I'm doing is I'm doing f of c1 into x2 minus x1, the length of this sub interval.

Similarly, go for the last sub interval at x2, you can see function is taking the minimum value. So this is f of x2. So f of x2 into this length will give me area of this sub rectangle. So what I'm doing is f of x2 into x3 minus x2.

So I concentrated in the sub intervals and I found the area of the rectangles and when I sum it up will I get the actual area under the curve answer is no because as you can see this area got missed this this and this area also got missed. So, this is not giving me the actual area under the cover. But then he gave a name to this number and what name did Riemann gave? He called this as LPF or some author also write as LFP.

L means what? The lower sum. Why lower sum?

Because we are looking for the point where function is taking the minimum value. So, this is called as the lower sum of a function for the given partition P. Okay or lower sum. sum for a partition P for given function f.

Usually I prefer this notation because first you should know function and then you partition right so yeah it's depend doesn't matter you can use any notation. So lower sum for given function function for a given partition P and this is called as my lower sum. So whenever someone talks about lower sum, that means you have to understand that over the sub interval he or she is looking for the point where function is taking the minimum value. So okay lower sum is not giving me the actual area under the curve.

So, let's go for the other side other side as in let's try to find the upper sum. That means let's try to look at the points where function is taking maximum value and let's try to see whether that will give me the area under the curve. What do you think? Let's check it out.

So now let's go for the upper sum. So now in this first sub interval you look for the point where function is taking the maximum value. So as you can see at here the function is taking the maximum value. So let me call this as a c1. So if I do f of c1 into the length of the sub interval x1 minus x0.

So what I will get this is the maximum value. So this is my length. So and this is my height. So, what will I get? I get area of this sub rectangle.

Do the same thing for the next sub interval for x1 to x2 at which point function is taking maximum value. You can see at x1 it takes the maximum value. So, therefore, So like this is the area of the sub rectangle the length into the height and for x2 to x3 at x3 takes the maximum value so if I do this so this is the height and this is the length so I get area of this sub rectangle.

So what I'm doing is plus in the next sub rectangle it was at x1. So f of x1 into x2 minus x1 plus here it was at x3. So f of x3 into x3 minus x2. So this is the area of the sub rectangles and we are summing it up. And since we are taking the maximum value of the function this sum is called as u.

fp upper sum for a given function over a partition p. So is ufp the actual area under the curve? So lfp was not giving me the area under the curve. The area got missed.

What about ufp? Is it also giving you the area under the curve? No. Why it is not giving you area under the curve?

You can see you are getting extra area. You are getting the extra area. So even ufp is not helping me. Ok, so neither LFP nor UFP is helping me. Okay, and what is the connection between them you can see your LFP is obviously less equal UFP because L is the lower sum so there the area got missed and U is the upper sum you are getting extra area.

So obviously this area is bigger than the this area. So LFP is less equal UFP. Okay, so that was one connection we saw between the lower sum and the upper sum.

And we also saw that neither lower sum nor upper sum is giving me the actual area under the curve. So what to do? Then comes the refinement into the picture. So let's try to see what is the concept of refinement and how does it affect these areas.

Okay, so we saw that neither lower sum nor upper sum was giving you or is giving you the actual area under the curve. So now the question is what to do? So Riemann said okay you refine this partition. Refine as in what you insert some more nodes into this partition. So let me call my new partition as q.

So what I'm doing so x naught. So let me rename this for simplicity. So let me call this as x1 x2. Let me call somewhere here x3 x4 or some or let me call this as x5.

Okay, so my x naught is here. Then x1, x2, x3, x4 and x5 is my right hand point. So what is my Q?

Q is also the partition of the interval a, b. And what can you say about p and q? One can observe that p is a subset of q because q contains p.

And since q has more nodes, that means what Q is doing, it is making intervals much more smaller. that means called as a refinement. So therefore, our Q is called as refinement of P refinement of partition P.

Okay, so this is what your Q is. Okay, now let's try to find upper and lower sum. Okay, so let me take a different chalk. Okay.

Now do the same thing. You concentrate on the interval. Okay, so make sure the color is different. Okay, so the purple one was the area for the partition.

and P now we are going to find the lower sum but for Q partition and let's try to see what is the connection okay now this is my first sub interval x0 to x1 at which point function takes minimum value at x0 function is taking the minimum value so this is my breadth and this is my length so this is the area I get okay so it is let me write down with the white chalk so it is f of x0 into x1 minus x0 Let's go to the next sub interval x1 to x2 at which point function is taking minimum value at x2 it takes a minimum value right this much is the curve. So at x2 function is taking the minimum value. So yeah, this is my height and the length so I get this area. So what I have is f of x2 into x2 minus x1.

Now you compare earlier when we found the lower sum for P partition we got this much as the area then we did the refinement and now if you find the lower sum we are getting If we are getting this much area and this much extra area, which is a good thing, right? So with Q I'm getting much more extra lower sum. I mean, I'm coming towards the closure of the area under the curve.

Okay, so let's do the same thing over here. If you concentrate on the interval x2 to x3. So again let me call this as say d1 at d1 it is taking minimum value so this is my height into breadth so this is the area of the rectangle then x3 to x4 at x3 function takes minimum value so this is the height this is the length so I will get this as the area of the sub rectangle here also if you observe initially I only got this much as the area okay initially I only got this much as the area But after refinement, I'm getting much more extra area. And the same thing works here.

Okay, so here, I have not inserted any node. So yeah, here it remains the same. If you want to insert not an issue.

Let me call this as x five and x six. So x five, and let me call x six as my b. So here if this is my node so this is what I have so over here this will be my lower sum because at x4 function is taking minimum value and here at x5 function takes the minimum value so this is the area I have so what you can observe so when I sum it up plus this area plus this area plus this area plus this area and when I add it up f of x0 into x1 minus x0 f of x2 into x2 minus x1 here f of d1 into x3 minus x2 here f of x3 into x4 minus x3 here f of x4 into x5 minus x4 and here f of x5 into x6 minus x5 and when you add it up this is my lower sum for the function for the partition Q. So, this is the lower sum for the partition Q and you can see that this Q is giving me the area which I am getting through Q partition is more than the area which I got for the P partition.

So, the relation which you can get between P and Q this is the first relation and another relation is your lower sum. For the function for the given partition P is less equal the lower sum for the function for the partition Q where Q is what it is the refinement of P and as you can see with Q and getting the extra area. So there's a good thing right so can keep this one in equality in mind, but again the same thing this is not giving me the actual area under the curve. So what should you do now? So, the idea is you further subdivide this interval into further smaller sub intervals.

That means you increase the number of nodes. So first you had three then you had six then you make suppose hundred nodes. What do you think your lower sum will keep on increasing.

Your lower sum will keep on increasing. What about the upper sum? So let's try to see the behavior of upper sum for the partition Q. Okay so we saw that after refinement Even Lpq sorry Lfq was not giving me the actual area under the curve.

Let's try to see the behavior of Ufq means upper sum for the function over the partition q. Now here also you do the same thing. So let me take a different chalk.

Okay so it's a blue color. Over the first sub interval look for the point where function is taking the maximum value. So here you can see at x1 is taking the maximum value. So this is the height.

Or okay let me. do this some nice partition so let me call I think my x1 was here if I am not wrong. Okay so you do the same thing now for this interval you look for the point where function is taking the maximum value so here you can see let me call this as d1 at d1 it is taking the maximum value so the maximum value into the height so I get the area of this sub rectangle. So what I'm doing is f of d1 into the length x1 minus x naught.

Cool, go for the next sub interval x1 to x2. So at this x1 I'm getting the maximum value. So this is the height into the length. So what I'm doing is I'm doing f of x1 into x2 minus x1 and now you see the magic the area got decreased after the refinement obviously u fp was not giving me the actual area under the curve I was getting larger area but after refinement upper sum got decreased okay let's try to do the see the same thing in the next sub interval for x2 to x3 at x2 it takes a maximum value so this is the area of the sub rectangle length into the height into the length and then from here x3 to x4 okay This is the area of the rectangle I get.

Here also if you observe little bit of area got missed. Okay same thing x4 to x5 here is the maximum value. So maximum into the length will give me area of this sub rectangle and from x5 to x6 maximum into the length will give me area of this sub rectangle and here also you can see this area got missed. So when I add everything plus this area this area this area and this area this i called as u f cube upper sum of a given function for over the partition cube okay so what good thing is happening and let me write down one more inequality over here i don't need this so what we know that your you your LFP is less equal LFQ LFQ is obviously less equal UFQ and UFQ is less equal UFP Right, this is what we have.

So, you had a curve over here, you wanted to find the area under the curve. What you do? You do the partition.

For the partition you find the lower sum area of the sub rectangles, you find the upper sum area of the sub rectangles. Those were not the area under the curve. So, you do the refinement. Lower sum increases. So, this is the curve over here.

So, upper sum, sorry, lower sum increases and upper sum decreases. That we are still not giving you the actual area under the curve. So you again do the refinement. Then again the lower sum will increase upper sum will decrease.

You again do the refinement. You keep on doing this. So when you keep on doing the refinement you increase the number of nodes.

Then your lower sum will keep on increasing. Your upper sum will keep on decreasing. And when they match. then that is actually the area under the curve. Okay, so that's what the idea Riemann gave.

So when first you had six three nodes, then six nodes, then you put 100 nodes, then you put 1000 nodes, then you put 1 million nodes, you put one crore nodes, you put 1 million zillion of nodes, so keep on adding the nodes. That means you make the length of the sub interval smaller and smaller, you make the length of the sub intervals smaller and smaller. And actually that there is one more node for this one more notion called is the mesh of P mesh means what the I mean the distance between the nodes become smaller and smaller.

So P is the partition mesh of P means what the distance between the nodes should get smaller and smaller. So keep on inserting the nodes. Keep on inserting the nodes.

that means the length of the sub intervals getting smaller and smaller and when that happens that means your n is going towards infinity that means so when your lower sum limiting case of the lower sum is equal to the limiting case of the upper sum and when they are equal we say that the function is remain integrable okay so what i'm trying to say over here is so what you had you had l fp less equal L F Q suppose R is the refinement of Q this is less equal L F R and you keep on doing that this will be less equal U F R. less equal u fq less equal u fp that means what as you keep on doing the refinement your lower sum will keep on increasing your upper sum will keep on decreasing and when they matches we say that the function is integrable and whatever that matching value is that is nothing but integration a to b f of x dx okay so let's try to understand this using geogebra software so when you go to this geogebra software at the i mean here you can see this search classroom resources so if you simply type remand sums i mean many good people have just made their activity public so that everyone can see so you can take whichever you like okay so i mean there are plenty of them so let's take let's go for this one suppose so now here you can see here is a function given to you here they have taken n 10 nodes n is equal to 10 means 10 nodes you can see there are 10 sub intervals you can play with this function and now here you can see as i increase the node see the pink one is the lower sum and the blue dark the blue one is the upper sum so you can see that upper sum is giving you more area lower sum is giving you lesser area okay this gray part missed away when you increase the nodes you can see the upper sum gets decreases and the lower sum increases and that means this rectangles are coming closer and closer to the curve so you can see if i try to zoom but still there is a gap okay so you can see there is a gap so you again increase the partition you keep on increasing like here there is only till hundred but yeah as you can see there is still gap so to further increase the node so you make 1000 then again there will be a gap. So, you keep on increasing the nodes your lower sum will come closer and closer to this curve and the upper sum will again come closer and closer to the curve and that's what I tried to tell you earlier. So, you keep on increasing the nodes your upper sum will come closer and closer to the lower sum. Now, you can see that the rectangles are very close to the curve.

They are not equal to because when you zoom in or zoom out whatever. So, you can see that there is a gap. okay so for smaller nodes you can see gap properly with your naked eye when you increase the nodes you need to zoom to see that actually yes there is a gap so you further increase the nodes and like that when so keep on increasing that means a norm goes to zero or your n goes to infinity if the lower sum is limiting case of the lower sum is equal to the limiting case of the upper sum you say that your function is an integrable function. here the good thing is you can click here and you can play with any function which you want we played with x square so you enter x square this is you get the graph of x square if you want sin x you get the graph of sin x you can play with sin x as well so you keep on changing the function and you can play with it so that's the good thing about this okay so now let's see how to write down the things properly okay so last part of the lecture how to write it down okay so you have a function from close a b to r the first thing is you first partition your domain so let p equal to x naught the left hand point x1 x2 up to xn the right hand point let p be a partition of the interval a b and your x i's are called as the nodes of the partition p not writing that statement but yeah you should write it down okay so once you have the partition what is the next thing you do you find the lower sum so how you find lfp you It is the summation right? You are summing up the areas.

So you are multiplying length of the sub interval with minimum value of the function where i is going from one to n because over each sub interval you are finding the area and you are summing it up. And what was your f of ci? Where f of ci is equal to what you are doing you are having a sub interval xi minus 1 comma xi over that sub interval you're looking the values of x and you're taking the functional values so f of x where x over is x is over this sub interval and you're taking the minimum of that right so over that sub interval you look for the point where function is taking the minimum value so you're taking the minimum of the functional values over that sub interval so that is what my f of ci is so that how one should write okay Then comes the UFP. What is UFP?

It is summation i from 1 to n f of z di into xi minus xi minus 1. And now here where my f of di is what? It will be the maximum of the function f of x where x is in this sub interval. So that is what my LFP and UFP is. So you had the lower rectangles, you submit up that is my LFP, you have the bigger rectangles, you submit up that is my UFP. But as we saw, they were not giving us the area under the curve.

So we did the refinement that means what we inserted more and more nodes. And when we did that, lower sum increased, upper sum decreased, you further keep on doing that you can keep on increasing the nodes, lower sum will keep on increasing, upper sum will keep on decreasing and they should match. Okay, that means what?

you are taking n going to infinity you are keeping on doing this process. So, so when you keep on doing some process, then the limit comes into the picture. And therefore, now let me call my lf let me call lf as limit n going to infinity of lfp that means you take n going to infinity. But I would prefer to write norm of P or the mesh of P goes to zero. That means what like the initial you had a B interval whose length is B minus a then you inserted nodes so the length became smaller.

You further insert the node so again the lengths become smaller. If again enter the nodes again the length becomes smaller that means the length should go to zero. Okay, I prefer this notation not this. Why so I will tell you maybe sometime in the next lecture but yeah anyways at least for engineering background I will say this is sufficient even if you write this okay but if you are doing pure mathematics then make sure this point is clear okay. Now what is my uf it is again same thing limit n going to infinity or you should write mesh of p going to 0 u f p and what I said at the end they should match that means if your LF is equal to UF we say the function is integrable that means we are getting the actual area under the curve okay so let me call this as number whatever the number they are same so this number is nothing but this value integration a to b f of x dx So that's how you find the integration.

Okay, so at the start, I told you that integration and derivative are different. And now you can see derivative is a slope of a tangent line, and integration is an infinite sum, right? Because this was a finite sum.

And then you're taking n going to infinity. So this is what is integration, it is an infinite sum, you're adding infinitely many numbers, and it is giving you a value. So integration is an infinite sum derivative is a slope of a tangent line. So you can see they're totally different. Okay, that's one thing.

And that's how this also comes up like this integration is nothing but actually s only. So I mean, the s is nothing but the sum. So integration is nothing but the sum. And they just elongated this s. So when you elongate the s, you get the symbol of integration.

So that's how this this represent the sum. Okay, this dx means what you're doing it into the small small, small, so x you are doing it dividing into the small small interval that means the change in x. So therefore this is the functional value f of ci f of di and this was the length of the sub interval and you are taking the infinite sum. So this is how this notion or this notation came into the world or came into the picture. So yeah so that's all about the idea of Riemann.

Now one can have many questions. I mean how to solve this by taking a proper example. So in the next lecture, I will take a proper example. We will solve that integration using this method, this algorithm, then we will also see example that which are not integrable, okay, not all functions are integrable, there are functions which are not Riemann integrable. So we will see that.

And we will also see some examples on Riemann sum, lower sum and upper sum. Okay, so yeah, so when this number exists, we say the function is Riemann integrable, or the integration of f exist, this thing. Okay, so that's all about the idea of Riemann.

If you have any doubt, you can ask me in the comment section and if everything is clear, do not forget to like, share and subscribe. Thank you.

Transcript for:Understanding Riemann Integration Concepts

Transcript for:
Understanding Riemann Integration Concepts