 So what I am going to do is, I am going to straight away start with some problems. Are you all ready with your notebooks? Yes. All right. So here comes, my friends, the first problem. Find the area, find the area lying above, lying above x axis and included between x square plus y square is equal to 8x and y square is equal to 4x. Question is clear to all of you. Find the area lying above the x axis. Mark these points, very important. And included between these two curves. Start solving this. And don't look at your screen till you have solved it. Because in the meanwhile, I will be also drawing some graphs. Okay. All right. So now you can look at the diagram at least. Is this the diagram that you have got? And is this the shaded area that we are talking about in this particular question? Yeah, yeah. Cool. Now a couple of things first. We need to identify the coordinates of P. So in order to get the coordinates of P, we need to simultaneously solve. We need to simultaneously solve the equation of the circle and parabola. Right? So all I can do is I can simply replace my y square as 4x into this particular equation. So I will get x square plus 4x is equal to 8x. Which means x square is equal to 8x minus 4x which is going to be 4x. That gives me 2 value of x which is 0 and 4. Which is quite obvious because they meet at 0, 0. And they will meet at a point P whose x coordinate will be 4. Y coordinate will also be 4. However, y coordinate is not of interest to me. I am not interested in the y coordinate. I am only interested in the x coordinate. Right? Now, again there are two ways to solve the problem. One is by taking vertical strips. Other is by taking horizontal strips. Right? Yes or no? Now let us solve it by taking, let's take vertical strips. Vertical strips. Okay? So let me make a green vertical strip like this. So I would need two types of vertical strips. That means one is like this and another would be like this. So there is a need to change the definition of the strip. Let me name it. It is a type 1 strip. This is a type 1 strip. This is type 2 strip. Okay? Now for type 1 strip, can somebody tell me the expression that I would be going to write to evaluate the area given by this particular zone. Given by this particular zone. This entire zone. Who will tell me the expression? Where is the, where is the upper part of the strip? The upper part of the strip is on the parabola, right? Now parabola is basically y square is equal to 4x. So there could be two possibilities that is y is plus minus 2 root x. Which one will I take with the plus sign or with a minus sign and y? Plus sign or plus? Yes, exactly. We will take the plus sign because I am dealing with positive y's. So I would integrate plus under root 2x from 0 to 4. That is from this point to this point. Correct? Yes or no? If you look at this strip, I am integrating upper minus lower with respect to x. What is upper here? It is a circle equation. Yes or no? Yeah. Right? But if I am little bit vigilant, I would require, I will see that you do not require integration for this. Because it is exactly quarter of a circle. Yes or no? Yes. What tells me it is a quarter of a circle? Very simple. If you see, this line is going to be perpendicular because it is also 4,4. This is 4,0. This is going to become a quarter of a circle. So frankly speaking, I do not need integration here. Okay? So I can directly say 1 fourth of pi r square. Yes or no? Which is going to be 4 pi? Yeah, 4 pi. Okay. Which integrated the normal way that you do? So 4 pi plus 2 times x to the power 3 by 2 divided by 3 by 2 will make it 2 by 3 from 4 to 0. Which will make it 4 pi plus 4 by 3 into 4 to the power 3 by 2 minus 0. Which is just going to give you? 4 by 3 into 8. Yes. It is just going to give you 4 root 4 into 4 which is going to be 16 into 232. Correct? 4 pi plus 32 by 3 square units. It is advisable to write square units because you are indicating the area. Make sense? Is that clear guys? All right. Now let us move on to another problem. Can I raise the screen now? Next is find the area bounded by, find the area bounded by y is equal to x cube minus x and y is equal to x square plus x. Please try this out. Again please do not look at the screen till you have made the graphs of these corresponding curves. All right. Now you can look at the screen. So basically I have drawn this as y is equal to x square plus x. Now I will tell you a way to draw it. It is basically you just have to notice that it is a parabola which opens upwards. Which opens upwards and secondly you just have to see where does it cut the x axis. So this cuts the x axis at 0 and minus 1. So if you factorize it and equate it to 0 it will cut the x axis at 0 and minus 1. Right? So in light of these two observations you can easily plot the parabola as you can see in red. All right? And how do you plot the cubic one? For the cubic one it is pretty simple see when you say y is equal to x cube minus x. Always remember a odd degree polynomial, a odd degree polynomial graph will always be starting from below and you know taking some turns and ending up. This is always a nature of a odd degree polynomial. A biggest example would be y equal to x cube. Right? It goes like this. Yes Sadaam. If the coefficient of x cube which is this coefficient if it is positive then it would be the other way round it would be like this. If the coefficient of x cube is negative. So these are some things which you need to keep in your mind when you are drawing the graph of a cubic polynomial. Is that okay? You could also know the sign by making your wavy curve. So that's doing the same thing actually. Now what are you doing? Any questions? I just had a doubt with the x cube minus x graph. Okay. I will do it on the GeoGebra as well. Okay. And where does it meet the x axis? So we simply equate x cube minus x to 0. Okay. And you can see that it can be factorized as x square minus x. And it can be further factorized as x plus 1 and x minus 1 equal to 0. So this gives you three points of intersection which is 1 minus 1 and 1. So as you can see here minus 1, 0 and 1. Is this clear so far to all of you? Now which area am I talking about? I am talking about the area which I have shaded in yellow. Okay. So let me just make some space for myself. Okay. Now I would need the coordinates of lot of points. First of all I would need the coordinates of this point. This point. Let me call this as point P. What would be the coordinate of that point B? Very simple. I need to simultaneously solve the parabola equation and the cubic polynomial equation. That means I need to equate this to this. So what do I get by doing that? 0 minus 1 and 2. Two of them is already visible. What are they? Minus 1 and 0. Correct. So if you write it as a cubic polynomial equation that is x cube plus x square plus 2x. Sorry. Minus 2x. x cube minus x square minus 2x equal to 0. You would realize that two of the roots is already known to me which is minus 1 and 0, isn't it? Aren't these two the roots? Only one root is supposed to be known. Now sum of the roots is going to be 1. Sum of the roots is going to be 1. How? Sum of the roots is minus B by A, right? And what is minus B here? Minus B is minus of minus 1 by A. A is also 1. Do you remember that this is minus B? This is B term. The coefficient of x cube is A term. See, whenever you write a cubic equation, we normally write it like this, right? Ax cube Bx square plus Cx plus B equal to 0. So if alpha, beta, gamma are the roots, the sum of the roots is going to be negative of the coefficient of x square by coefficient of x cube. So the sum of the root is going to be 1. So we already have minus 1, 0. So what is required here to give me 1? What is this question mark? You will say 2. So this point is going to be x equal to 2. Is it clear? Yes or no? Yes. So let's be a little smart while we are trying to find out the points because we don't want to waste too much of time finding the points, okay? So let me just make some space for myself. Now, can somebody guide me how many definition of the strips I need to cover up the entire area? So one type of strip that I need is the one which is like this. Yes, and the other type of strip which I need is the one which is like this. So I need two types of strips. So the green one will help me to get this area as you can see on your screen and the black one is going to help me get this area, correct? Now what should I write for the area which is covered up by the green strip? Can we have the expression for that? Focus on the upper minus lower always. What is the curve on the upper part of this? Is it the parabola or the cubic one? Cubic. Correct. So cubic minus, what is the curve on the lower part of the strip? The parabola, correct? And this integration has to be done from where to where? Such strips would be required from which point of X to which point of X? Minus 1 to 0. Excellent, minus 1 to 0, very good. Plus, now focus on the black strips. Let me choose the color black, yeah. Now upper part of this strip, what is the equation? Parabola. Parabola, excellent. Minus lower part of this strip, what is the equation? The cubic. This I have to integrate from which value of X to which value? 0 to 2. Right? Now just take one minute and give me the answer for this. So just check your answer, your answer should come out to be 37 by 12 square units.