 Now moving on basic transformations. Is it clear what happens when you replace X with X minus H? What happens when you replace X with X plus H? Everything is clear Just to quickly recap in any function if you replace your X with X plus H where H is some positive quantity The graph is going to shift H units to the left right Let me just take this example Let me take this function Y. Okay So let's say Y is equal to I'll take another function Fx is equal to I take a Same function Same type of function X square Minus 5x Let's say plus 4 Yeah, this is my function Okay, all of you can see the function now Now in this function, I am going to replace. I am going to plot another function by replacing my X in the function of H with X plus 2. Do you see the red graph? What has happened? It has got shifted two units to the left correct Yes, in the same way if I replace my X with X minus 2 You would realize that the graph will now shift two units to the right direction correct In a similar way if you replace your Y with If you replace your Y with Y plus K Your graph is going to move up by K units if K is positive Okay, and if you replace your Y with Y minus K the graph is going to go down by K units Yes, I know Let me show you that in the same function. I'm not going to plot Yeah, so Y is equal to X square minus 5x Plus 4. Okay Now what I'm going to do is I'm going to change my Y with Y plus 2 see the graph going Will you go up or go down? Y plus 2 means the graph is going to go Down. Yes or no as you can see it has gone down Okay, if you replace your Y with Y plus K The graph is going to go down and if you're going to replace your Y with Y minus K The graph is going to go up Okay So as you can see the graph has gone down Now the same equation now I'm going to change Y with Y minus 2 see the graph going up. Is that fine? So now we are going to continue with The session that we have over here So this concept is Understood by all of you that when do we get a positive answer and if you follow upper minus lower and Integrate from a lower value of X to a higher value of X You will always get a positive answer. Is that clear? Now there are some area that you can actually calculate by considering Horizontal strips in the previous case I talked about vertical strips now There are some area which you can find only by taking Horizontal strips, let's say I want to find the area given by this figure Okay, and let me tell you these are two curves X equal to f of y and X equal to g of y another important thing here is that we need to write the equation of the curves as X equal to f of y and X equal to g of y in this case Unlike the previous case our equation was written as Y equal to f of x and y equal to g of x right so here you have to make X the subject of the formula I'll tell you why it is required So let us say you are integrating the area or you are finding the area Sandwiched between these two function Okay, between y equal to a till y equal to b So what you're going to do next is you are going to take at a distance of y a thin horizontal strip like this So you are going to take this trip. Yes or no? thickness is dy correct Now if I ask you What is the height of the strip? Height of the strip will be dy no doubt about it correct What is the? Width of the strip what is this distance? You would say it is nothing, but the difference of the X coordinates at these two points, so it is nothing but X2 minus X1 correct and If you look deeply what is X2 X2 is nothing but f y So X2 is nothing but f y X1 is nothing but G y yes or no So the difference in this is going to give you the width of the strip and which if you multiply with the height is Going to give you the differential area da right and now I'm going to integrate this from Y equal to a till y equal to b and Remember if you want your answer to be a positive one if you want this answer to come out to be a positive one Ensure that this is positive and This is positive Now can somebody tell me how do you ensure that? Fy minus g y is positive Any guesses, how do you ensure f y minus g y is positive? Fy should always be the curve. We should write on the right or left Right right exactly. It should write on the right and G y should be the curve. We should lie on the left because right minus left means you are Doing a higher value of X minus a lower value of X remember as we go right on the X axis We start getting positive values or increasing values of X and as we go to the left of the X axis We get decreasing value of X So right minus left will always be positive so if you do right minus left and integrate from a Lower value to a higher value You will always get a positive answer you would never need a modern all Is that clear? So guys in order to know whether my curve is on the right or left we again have to be very very good with our Graphs if graphs is not good. This chapter is not meant for you correct So graph and the transformations which I have discussed so far are the biggest prerequisite along with some understanding of functions In order to excel in this chapter Are you getting this point? now The first question that arises in our mind is When we are solving the graph when you're solving the question of area under curves What is the first thing that we should do? So let me point out the first thing first always plot the graph as Accurately as possible Plot the graph as accurately as possible This is the first step that you should do okay second step is Second step is decide whether you have to choose Vertical steps integrate or horizontal steps to integrate or you could use both This is the biggest call which you should take whether I can integrate the area by just using vertical steps Or just using horizontal steps or would I need both? That means How would my integration process be less if I break the area into strips Let me give you a very simple case to understand point number two Let us say I have a circle Okay, let's say the circle is y is equal to f of x equation and I have a parabola the parabola is Let's say y is equal to Y is equal to G of x okay, and I want to find out this area I want to find out this area Okay Now what kind of steps would be suitable to find out this area any guesses? Horizontal horizontal would be the most suitable one why for very simple reason is Your if you make a horizontal strip like this Let me choose some other color if you take a horizontal strip like this You are only going to use the curve on the right Which is going to be your now here the equation is not given to you in the form of x equal to F of y so first you need to convert this as x in x equal to some let's say h of y and This also you have to convert x equal to some k of y So this area could be easily found out if you could know the y coordinate of this point. Let's say a So you can integrate it or you can find the area easily by doing The x on the rightmost function, which is your h of y Minus k of y that is your curve on the left of the function left of the strip That means where does this lie and this lie so this lies on this and this lies on this Correct, so the difference of the x coordinate from 0 to a this would simply give you the answer in straight away one shot However, if somebody wants to solve the same question by taking vertical strips Let me tell you what challenges the person will face Let's say first of all he has to find this point, which is a this point, which is B okay and He would require to choose Two definitions of the strip one like this and Another one like this Correct You would need two definitions of the strips one is this fellow and another is this fellow Yes, I know So can you tell me what expression should he write? Assuming that the circle is y is equal to f of x and the parabola is y is equal to g of x He would write something like this Integral from 0 to a Now where is the tip? Where is the upper end of the strip? g of x Correct, where is the lower end of the strip at y equal to 0 plus Integral from a to b now, where is the upper end of the strip on the circle? Yes or no Circle is f of x minus 0 So unnecessarily you will have to take two integrals to solve this problem Whereas in this case you could have solved this question in just one integral correct So guys you have to be very very judicious in taking a call Whether you need a vertical strip or a horizontal strip or a combination of both in order to get your area Are you getting this point and those two and finally the last step is You would have to be very very careful in integration right Though the integrations involved will not be of very high level Because we are dealing with some standard functions like circle ellipse parabola hyperbola No, we'll be dealing with straight lines. We'll be dealing with modulus function We'll be dealing with gif So you have to be very very careful in performing the integration If these three steps are taken care of then you'll find this chapter super easy Okay