 we'll take few more problems and then we can end this session. Okay. So now let's take up some questions where you would realize that the difficulty of the question is not in the integration part but in finding out the function which is supposed to be integrated. Okay. Let's take a question like that. So in this chapter is there actually like any other formula or any other new concept or just like this? No, no, it is just based on the basic geometrical interpretation of definite integral and of course your knowledge of functions. Okay. Okay. So let's take this question. There is a function at all. So how tough can this chapter questions be set? A typical example of typical question of j can look like something which I am going to write next on your screen. Okay. So just see whether you are able to understand this. So there is a function f of x which satisfies this functional equation. Just mind the correction here. Supposed to be a minus. Okay. Find the area between f of x to the power of 2 by 3 plus f of y to the power of 2 by 3 equal to 12. f of y to the power of 2 by 3 equal to f of x to the power of 1 by 3. Is the question clear? Now here the challenge is in finding the function itself. So what is your f of x here? If you get that, here the challenge is in finding f of x. Correct? Sir. Can you hear me? I tried to solve a bit. Okay. And I figured out that my f of minus x was coming minus of f of x. Okay. So it's an odd function, right? Yeah. Okay. All right. So let's solve this and make an attempt to solve this. So let us substitute x plus y as x and x minus y as y. Let's see what happens to this expression. So the left hand side of the equation is going to become x f of y minus y f of x. And if you look at this expression very carefully, it is actually made up of 2x into 2y into x plus y x minus y. Correct? Right? Now what is 2x? 2x is nothing but capital X plus y. And 2y is capital X minus capital Y. Correct? This is nothing but x into y. So what did I do? I just transformed this entire functional equation by making this substitution as x plus y as capital X and x minus y as capital Y. Okay. Correct? Now if you simplify this, you get x square minus y square times xy equal to f of y minus y of f of x. Okay. Does it give you some hint? Does it give you some hint? If still if you haven't got any hint, you can actually multiply this and you can write it like this minus y cube minus x y cube minus of minus x cube y. Okay? Now if you compare this, if you compare this with this expression and you compare this with this expression, you would realize that x into f of y is minus x y cube, which means f of y is actually minus y cube, which means your function is nothing but minus x cube. Right? So if this is the finding that you can get from this entire functional equation, then your life will be pretty easy because then you can predict what are these functions. Right? So this is going to be x square plus y square equal to 12 and this is just going to be y square is equal to minus x. So basically you end up with a circle and a parabola which opens towards the left. So all you need to do now is to find out the area between this curve and this curve. So can we do that now? So do you need any explanation from this present page or should I go to the next page? Can I clear this off? Yes, sir. Yes, sir. So we are now going to find out the area between x square plus y square is equal to 12 and y square is equal to minus x. So basically we are going to get this circle and we are going to get this kind of a parabola. So which area do I need? I need this area. Now guys, while dealing with circle also please be very, very careful because when you say the circle equation is this, okay, it comprises of two equations. One is y is equal to under root of 12 minus x square and one is y is equal to minus under root 12 minus x square. So the upper part of the curve is y is equal to under root 12 minus x square and lower part of the curve is y is equal to minus under root 12 minus x square. So depending upon which side you are taking up for the circle, you have to change the equations like this. So if you are taking the upper part, your equation will be this. If you are taking the lower part, your equation will be this. So anyhow in this particular curve, it would be better if you deal with one side and double the area. So let's just find out this part first, okay and double the area. So what would be this point of intersection? Minus 3 comma root 3. Okay. So for that you just have to solve x square minus x equal to 12. Yes or no? So x square minus x minus 12 equal to 0. So is it factorizable? Yes. So x minus 4 plus 3 x minus 4 equal to 0. So I get x plus 3 and x minus 4 equal to 0. Now what is happening over here? What is happening over here? I am getting two values, x is minus 3 and x is 4. Which one will be accepted? This fellow. Minus 3. Okay. So this point will be minus 3 and what will be this point? Minus root 12. Correct? Now you tell me what should be the expression of the area that I should be writing? What is the expression for this area? If I take this differential strip, what is the expression that I should be writing? Minus root 12 to minus 3. Tell me. Under root of 12 minus x square. Correct? And if I take a differential strip like this, the expression that I would be writing is minus 3 to 0 under root of minus x. Now let me tell you many people are surprised by this. How can this happen under root of a negative quantity? Under root of a negative quantity. Please let me tell you this is not under root of a negative quantity because x itself is negative. Okay? So from minus 3 to 0, x itself is negative. So the integration of this will happen in a normal way that is minus x to the power of 3 by 2 divided by 2 by 3 with a negative sign. Correct? But technically how can you actually calculate an area like that? See again, why cannot we calculate? Why not we can calculate the area of this? Because then that comes under a complex number. No, no, it doesn't come because x itself is a negative quantity. So negative of a negative is actually a positive quantity. Okay. Just don't go by the presence of a negative sign. Okay, we have a negative of a negative means it is a positive number. Okay, understood. And which formula will I use over here? Integral of under root a square minus x square. Correct? And what is that answer? x by 2 under root a square minus x square plus a square by 2 sign inverse x y. Correct? So now you know these two results so you can find the answer out. So guys, what did we conclude here? The conclusion was the question was not difficult because of the integration. The question was difficult because it made your life difficult in finding the function out. So in J, most of the concepts that are asked would not be difficult with respect to the actual problem. It would be difficult with respect to the supporting concepts. Right? So therefore in the beginning of the chapter itself, we discussed that we have to be very good with our functions, graphs and their transformations. Correct? So with this, we complete this chapter. Congratulations to all of you. We finish off this chapter. Okay. So next time when we meet, we'll be announcing the topic so that you can join our session. Okay. Thank you so much for joining in all of you whoever is watching us over and out from Centrum Academy.