 All right, so we can start with the session today. So the topic for today's session would be area under curves. So you would have already done definite integrals. And this is going to be an application. It's going to be an application of definite integrals. So are you sharing screen? Yes. Is it not visible? It's not visible. Just give me a minute. All right, so we'll start with area under curves, which is actually an application of definite integrals. So definite integral is something that we have already studied in our previous chapter. And this is going to be a chapter which is going to be based on the following prerequisites, your understanding of, first of all, the geometrical interpretation. The geometrical interpretation of integral f of x with respect to x from a to b. So this chapter is primarily based on how do you understand the geometrical meaning of integral f of x from a to b. Secondly, the biggest prerequisite is you should be aware of functions along with the graphs. And needless to say, you need to know the transformations of these graphs as well, transformations of graphs. So when I talk about geometrical interpretation of f of x from a to b, what did we study about it in the chapter, definite integrals? Can anybody recall that? It is the area under the curve. Right. Area under the curve, which curve? f of x. Till? From a to b. All right. So basically when we talked about, let's say this is my function f of x. OK. And if you are integrating this function from x equal to a till x equal to b, OK? When you say integral of f of x from a to b, it actually means that you are trying to create at a distance of x a very, very thin rectangular strip. Right? So if you make a very, very thin rectangular strip like this, OK, at a distance of x, having a thickness of dx, OK? Then what would be the height of this rectangular strip? You would say the height of this rectangular strip is actually given by this term. This term gives you the height of the rectangular strip. Correct? And what does dx give you? dx actually gives you the thickness of this strip. Correct? So the area of this strip is going to be height into thickness, which actually gives you the area of this rectangular strip. And when you integrate this area, which I call as da, from a to b, you end up getting the area of the curve right from x equal to a till you reach x equal to b. Isn't it? So a few things that we need to understand over here. If you analyze this pretty deeply, you are actually writing the height of this strip as the difference of the y-coordinate here and the y-coordinate here, OK? Just focus on the dots which are made on the screen, OK? Let me use another color, so this dot and this dot. So your height is basically the difference in the y-coordinate. Let me call this as y2, and let me call this as y1. So height is actually y2 minus y1, correct? Now if I ask you, what is the y-coordinate? What is the y-coordinate of this point? I mean, what is the value of y2? You would say y2 value is nothing but the value of the function at x, because to reach to this point, if I travel x from the y-axis and go up directly, I can see the y-coordinate at that point, correct? In a similar way, if I ask you, what is your y1? This is your y1, by the way. What is your y1? You would say y1 would be 0, because you are on the x-axis. x-axis equation is y equal to 0, correct? So the term here that you have written is actually nothing but f of x minus 0, which is nothing but the difference of the y-coordinates of the top of the strip and the bottom of the strip, correct? And of course, this dx is going to be your thickness, right? So you can call the thickness is going to be dx, OK? And when you multiply these two, you get something called the differential area. We get the differential area. So normally, when we do this, you are actually getting the differential area. And the process of integration is nothing but a process of summation, correct? Integration is nothing but summation of continuous quantities or continuously varying quantities, OK? Yes or no? So area under the curve is going to be the area which is obtained by adding all these thin strips right from x equal to a till x equal to b. Many times, what happens while we were doing definite integrals? We used to get negative answers, right? Is this clear so far, what has been told so far? So basically, when you're talking about integrating f of x, it is actually trying to find out the area between the function and the x-axis trapped between two vertical lines, x equal to a and x equal to b. Any questions so far? No, sir. OK. Now what I'm going to do next is I'm going to extend this to finding the area between any two functions. Let's say I have a function y is equal to f of x, OK? And I have another function y is equal to g of x, OK? And let's say I want to find out the area between these two functions from x equal to a to x equal to b, OK? So I'm looking for this area. Now the problem here is that if I do the integration of f of x from a to b, will I get a positive answer or a negative answer? Now there's a big debate about the positivity and the negativity of the area while dealing with area under curves. So I would like to highlight here a few important things which you should all keep in mind while you're trying to find the area under the curves. If you follow these following steps, then you would never have to take a mod in your answer because many a times some books say that if you are getting a negative answer, mod it. But if you follow the steps which I'm going to share with you right now, you would never need to mod any answer. You will automatically get a positive answer. Though I've already told this to you in the definite integral class, I would still repeat this again over here. If you are taking the height of the steps in such a way that you choose the y of the upper curve, minus y of the lower curve, you would always get this height as a positive height. Secondly, if you integrate from lower value of x, if you integrate from lower value or you can say like this, if A is less than B, that is your lower limit of integration is lesser than the upper limit of integration, then basically you are incrementing in x. If you're incrementing in x means your dx will automatically become a positive quantity. So if your change in height and in change in x, both are positive quantities, then your differential area of the strip will also be a positive quantity. And your integration from A to B will always result into a positive area. This will always be a positive area. So you would never need to take a mod while finding the area under the curves. Is that clear? So now, if you look at this present scenario, if I say what is integral f of x from A to B, will it be positive or negative? Will it be positive or negative? It will be positive, and what is the reason for that? If you see it is actually integral of A to B f of x minus zero, right? Yeah, yeah. Correct. So is f of x above or y equal to zero above? F of x. F of x will above, right? So this is your upper curve, this is your lower curve. So upper minus lower will always give you a positive answer. Are you integrating from a lower value of x to a higher value of x? So d of x will also be a positive quantity. So this will be a positive quantity and this will be a positive quantity. And therefore, we say that this area over here, which I'm shading in green, that would be a positive area. Yes or no? Yes, sir. In a similar way, in a similar way, if I ask you, let me just make some space for myself. In a similar way, if I ask you, what is the integral A to B g of x dx? Will it be a positive area or will it be a negative area? Now you would say if I integrate g of x from A to B, I'm actually writing this expression. Correct? Yeah. And unfortunately, your g of x would be the lower curve and y equal to zero will be your upper curve, isn't it? As you can see, g of x lies below y equal to zero. A x axis is y equal to zero. So this will be a negative quantity. And since you're integrating from a lower value of A to a higher value of A, this would be a positive quantity. Right? Yes. As a result, you would get a negative answer. Right? So this answer, let me shade the area with red in color. This red area that you see would be a negative area. Okay? Now, the question demands me the entire area. So if I blindly add f of x plus g of x integral with respect to x from A to B, do you think I will do justification to the problem? Will I get my answer? Is this the area which I require positive? Plus, let's say this magnitude is 10 and this magnitude is two. If I add this, my answer would come out to be eight, right? Yeah. I'm doing the algebraic sum of the area over here. However, my desired answer was 12, right? So my desired answer is 12, so they are not equal. So doing such activity will not fetch me the answer. Are you getting this point? Instead, what should I be doing? What should I be doing instead? Subtract it. Subtract it, correct? So instead you would say as you rightly pointed out, my actual area would be in this case, integral of f of x minus g of x. Yes or no? Yes or no? If you go a little step back to be more precise, you are actually going to do this. Integral of f of x with respect to x minus integral of g of x with respect to x. Isn't it? If you combine this, you're going to get this, right? Now, in actual sense, what are you doing here? In actual sense, you are actually doing upper minus lower again, right? If you take a strip like this, if you take a very, very thin strip like this, okay? Where is the upper end of this strip? Upper end of this strip is lying with f of x. Lower end of this strip is lying with g of x. So you are actually doing f of x minus g of x in this case. Yes or no? Yes. Okay. So while dealing with area under the curves, if you really want to get to the accurate answer, you must know which is the upper curve and which is the lower curve. And for this, you should be knowing the graphs of these functions. So the biggest prerequisite, the biggest prerequisite for solving questions based on area under curves is knowing the graph of these functions. Correct? Now, just a quick revision of the graphs. Surely you would be knowing all the graphs very well. And I'll just quickly revise with you the transformation as well, okay? So let's look into the graphs of some functions. So all the graphs, I could easily draw on this particular GeoJabra app, okay? So is there any graph that you people feel that you have forgotten? You're all aware of the graph of quadratic equation? Yeah. Okay. Are you aware of the graph of the cubic equation? Yeah. Anybody who has forgotten the graph of logarithmic equation or exponential equation? Nobody? No, sir. Okay. Are you aware of the effect of modding the function or modding x or modding y? This part I need to revise, okay? Yeah. Okay, let's say I have a graph of a quadratic equation, which is x square minus, is my screen visible to all of you? Yes, sir. Now, in the same graph, if you mod the entire function, if you mod the entire function, what happens? This is something which is very well known to all of us. The part of the graph which was below the x-axis will now go above the x-axis. Like something like this happens? Yes or no? Yes, sir. Okay. If you just mod x, what will happen? So let's say I just mod x. If you just mod x, what will happen? The part of the graph, the part of the graph which was actually on the positive x-axis, this part, do you see? Originally, the graph was like this. If I plot the original graph, it was like this, right? So you remove this part. You remove this part. And whatever is on the positive x-axis, that gets reflected about the y-axis, right? So this is the effect of modding the graph that whatever was on the positive x-axis, this part, it gets reflected about the y-axis like this. Like what you can see on your screen right now. Yes or no? Yes, sir. Okay. What is the effect of, what is the effect of modding y? Any guesses? So the part which below the x-axis gets reflected about, right? No. That is modding the entire function. I'm modding y now. I'm just modding y. All of you please pay attention over it. If you're just modding y, you see what happens with the graph? I'll tell you exactly what is happening over here. Earlier your graph was somewhat like this. Yes. Now when you're modding the graph with only y being modded, what happens? The part of the graph which is below the x-axis that is neglected. And whatever was above the x-axis, that gets reflected down like this. Are you getting this point? Whatever was below over here, that got removed. So we removed this part. Okay. And whatever was above got reflected below. The logic for this is pretty simple. See, when x-square minus 3x plus two was already a positive quantity. Let's say it was a value three. Okay. So when you say mod y is equal to three, y can take two values plus and minus three both. So for a value of three, for a value of x where you got y as three, for the same value you will get y as minus three as well. Okay. And why this part of the graph disappeared? Let me explain you that. If your x-square minus 3x plus two were negative, let's say negative one by four, right? As you can see if it is going below the x-axis it would be some negative value. Now when you say mod y is equal to negative one by four, is it possible? Is it possible? No, mod of any value. The y itself can be negative, but mod of y can never be negative. So therefore all those values of x where x-square minus 3x plus two became negative was removed from the domain of this function. Does it make sense? Yes or no? Yes, sir. All right.