 So, let us now also describe what is called other type of domains. So, domain of type 2 domains. So, you would have already guessed what is type 2. Instead of saying x is between some limits a and b and y is varying between something is other way around, y is fixed and x is going to vary. So, d is written as x, y, such that y is lie between some limits a c and d and for every such y, x lies between some function. So, let us call it as phi y and less than or equal to some other notation phi eta, it does not matter, we can write eta y itself does not matter. So, what does this look like? It will look like y lies between some c and d. So, let us write c and a is d. So, these are the lines in which and for every y, we have to look where does x go from. So, x is going to be horizontally, it is going to vary horizontally. So, here is how much for every y affects, how much I have to move along x axis and see how much remains inside the domain. So, it starts at somewhere phi y. So, let us write somewhere as, so this is phi y. So, it starts here and goes up to where, goes up to some function of eta y. So, it goes up to some function, so let us say this one, so that is eta y. So, this is for any point y, so that is a limits, where phi y and eta y are functions. Let us assume they are continuous, so that no problem comes, continuous functions defined on c to d. So, in type 1, x, so vertical things are fixed, x lies between a and b and for every x, we are going to move vertically, how much you move from bottom to the top limit. In type 2, your y is going to be fixed between two limits c and d and for every y, you are going to move horizontally, so that you are inside the domain. So, let us look at some previous examples and analyze them. For example, can I say that semicircular disk is type 2 means what? y lies between something and something. So, obviously, y lies between 0 and 1. To be inside the domain moving horizontally, when you move horizontally, you will be going from here to here. But that is the same function, the limits do not change. So, it is not of domain of type 2, semicircular disk, upper semicircular disk is of type 1, but it is not of type 2 because I can say y lies between something. But for every y, when I move horizontally, it does not say it goes from some lower limit to upper limit. It is the same function it says. So, it is not of type 2. Let us look at the complete circuit. This one, complete disk, is it of type 2? It was of type 1. We saw it. So, to be saying it is of type 2, let us write d is equal to all x, y. Where does y go from? Again goes from minus 1 to plus 1. How does x vary? Minus 1 to plus 1, how much is the x varying? It goes from the left side to the right side. Can I describe this boundary as a function of y? This as a function of y. So, x, it starts here on the left side. So, what is the equation of the left side? y, for every y fix, what is x? So, x is equal to 1 minus y square with a negative sign. Is that okay? That is this part. So, this part is, equation is minus 1 minus y square square root. And what is the right side? What is this side? That is with a positive sign, square root of 1 minus y square. Is that okay? I have to write down the left side as a function of y, because what is the domain of type 2? Function of y, function of y. For every y fix, what is phi y? So, x goes from, so this is of type, so d is of type 2 also. So, this is of both type 1 and type 2, both. You cannot write this as, because this equation of this, for every y, where does it go from? It goes from this equation to the same equation as a function of y. You will have a problem. So, that is the reason. Let me look at this example. This was of type 1. Can I say this is of type 2 also or not? So, what is this domain d? x, y. Where does y go from 0 to 1? So, y goes from 0 to 1. Now, x, for any point y, I have to see how much x varies. So, x starts at this line, vertical line and goes up to, so it starts at x is equal to 0 and goes up to 2y. I have to write x as a function of y. This is eta y. It should be a function of y. So, this is also of type 1 as well as type 2. So, d is of type 2 also. Let us look at this one. This was a domain d of type 1. This was type 1, because we wrote it as it goes from 0 to x goes from 0 to 2 and vertically, we know it goes from one limit to another limit on equation 2. Can I say it is of type 2? No. Why not? So, y goes from 0 to 2 and if I any point, if I look at here, I have to go from here to here, but that depends on whether I am there or so the function upper limit changes depending upon the point y. Again, if I want, I can cut it into two parts as union of two non-overlapping domains of type 2. I can do that if I want to. Otherwise, domain type 1 is also fine. So, is it clear what is domain type 1 and type 2? Yes, type 1, x is fixed and y varies between limits and type 2, y is fixed and x varies between limits. So, depending on your convenience, you have to interpret the domain as type 1 or type 2 or cut it into parts so that it becomes a union of domains of type 1 or type 2. Now, the question comes why I am discussing all these domains of type 1 and type 2, because there is a theorem which helps us to compute double integral. This is the theorem. It is called Fubini's theorem. So, this one helps us to compute the double integral. So, it says, let f d contained in R2 to R, f d closed bounded, f double integrable on D. The basic idea of this theorem is very simple. So, let me just explain that in the picture first. So, let us suppose D is of type 1. So, that is the first assumption. In case D is of type 1, what does the picture look like? The picture will look like, so that is A, this is B and for every point in between x goes from here to here. So, let us give what was the name, we got it psi x. It does not matter actually what eta x. So, D is equal to all x, y such that x lies between A and B and y lies between eta x and psi x. That was domain type 1. It says, if it is of type 1, basically what we are looking at, see what we are looking at is, what is the volume above this domain of something. Now, if I fix a point x here, if I fix the point, then this is a line which is in the domain. This is a line which is in the domain where x is fixed. What is varying here? y is varying. So, all these points are x, y, where y is inside y goes from eta x to psi x. Now, when I want to raise it above, I can think of at every point what is the value of the function and integrate. So, I want to integrate over this line. So, what will that give me? If I integrate over this line, so integral of bottom is eta x, top is psi x, f is the function and what is fixed here? x. So, I am integrating with respect to y. On this line which is in the domain, x is fixed and y is varying from eta x to, on this line, my function is defined because the function is defined on the whole domain. So, if I look at this integral, what will this represent? That will represent, look at the graph of the function above this line. So, there will be some graph. It will give me the area below the graph of the function. So, this integral will give me the area below the graph of the function. So, you can think it off with this line as the base, there is a sheet, this is paper sheet, top is f of x, y, x is fixed, y is varying. Now, if I add up all these sheets, I should get the volume. If I add up, if this is a sheet, it is the area I know. What will be the volume? If I take a small thickness, that will be dx. I want to add up. So, I should integrate x is equal to a to b. That should give me the volume. Is that clear to everybody? Yes or no? Picture, this is my domain. Let me probably, if I can draw three dimensional pictures, let me go back to my original picture and see whether I can add something there. In this, this is my surface at the top. I think I can, I can extrapolate the surface. So, this is my surface. If you are, if you are fixing, what are we fixing in this? We are fixing x and letting y vary. So, let us fix an x. In this, let us fix an x, some color which is good. So, let us fix an x. So, this x is fixed. So, I am looking at this line. Sorry, is it okay? That is a line, x is fixed, y is varying. Is it okay? This side is my lower limit. This side is my upper limit for the domain. When I integrate, so as x varies in this, where it is, the function value is going to vary. So, they are going to vary like this on the surface. So, these are the values f, x is fixed, y is varying. x is fixed. For every point, there will be a height. For every point, there will be a height. For every point, there will be a height. So, that will be a curve. There will be a function. For that function, x is fixed, y is varying. So, what happens? When I integrate this, what should I get? When I integrate, I should get the area of that sheet. I should get the area of that sheet. And as x varies, I will get the different sheets, add up all these sheets of small thickness. We will get the volume. If you like, you can think of a book. What is the volume of the book? The volume of each page added together. That is the simplest way of looking at it. Fibonacci theorem says the following. Suppose D is of type 1, then this quantity exists, meaning what? That means it consists of looking at the function f x, y. So, it exists meaning what? For every x fixed, integral eta x to psi x of f x, y, d y exists. This integral exists and further. So, f x, y, x was fixed. So, what does this integral depend? What is the value of this integral? That will be a number, which will depend upon x. Now, I sum it up. Sum it over x. Further, this is integrable with respect to x. It is also integrable with respect to x. So, x going from A to B, we are saying this is integrable. So, eta x, psi x, f x, y, d y, that is the area of the sheet into small thickness d x. That gives the volume of the thin sheet added up together. That is same as the volume that we are starting with. So, the double integral of a function over a closed bounded domain, if the function is integrable, if the domain is of type 1, then the double integral can be computed by looking at function f x, y, one variable at a time. Here, we are fixing our x. x is fixed here as a function of y. Integrate one variable and then integrate this as. So, that gives you another function of one variable. That is also integrable and that integral is same as the double integral. So, this is what is Fubini's theorem for domains of type 1. If your domain is of type 1, then you can calculate the double integral as one variable at a time. So, the left hand side, these two integrals are called iterated integrals. These are called iterated integrals. You are iterating the double integral by one variable. These are iterations. So, that is why each one, the inner integral is called iterated integral with respect to y and that is integrated. You get the second iterated integral. So, Fubini's theorem says if your domain is nice type 1, then double integral can be computed as iterated integrals. There is type 1. Similarly, there should be type 2. So, let us write type 2 also so that we understand again. So, second, if D is of type 2, so what is type 2? So, D is equal to x, y. So, that y is between some c and d and for every y, x lies between some function say phi y and psi y, where phi and psi are nice continuous functions. One can relax these conditions, but let us assume they are nice. This is of type 2. Then, let me write the integral of fxy dx. I am fixing y. Now, we are fixing y. So, it is between phi y and psi y. This exists, is integrable with respect to y. When I integrate this with respect to y, so phi y, psi y, f of x, y dx. Integrate this. y was fixed. So, this is dependent on y. So, dy integral y goes from c to d exists and is equal to the double integral f of x, y. Then again, it is equal to the double integral. So, computation of double integral by Fubini's theorem becomes easier in the sense that you can push the problem to one variable at a time depending upon whether your domain is of type 1 or of type 2 or you can cut it into pieces of type 1 and type 2. So, basic fundamental things are building blocks are integrals over domain of type 1, domains of type 2. Then everything is nice. You can compute the integrals. So, this is a computational aspect of Fubini's theorem. So, probably let us look at some examples. So, whatever I have said, this is type 1. X goes from A to B. Vertical goes from one function to another function. So, let us just revise anyway. Similarly, type 2 will look like this. Horizontally, there are limits c to d and for every moving along the horizontal line, it goes from one function to another. So, it goes from this function to this function. That is type 2. So, example of the disk and so on, we have looked at. So, circular disk is both of type 1, type 2 and Fubini's theorem says that if it is elementary region of type 1, then the double integral looks like this. Integral, x goes type 1, x goes from A to B. The iterated integral is f x y phi 1 x 2 and phi 2 of x d y. So, integrate with respect to y, with x fix, whichever is the finite limits c to d or A to B, that goes out. That is the outside one. Inner one is the variable thing that you... So, that is type 1. So, that is what... If that picture makes sense, you are looking at the sheet and the sheet is moving. Similarly, of when it is type 2, y is between c and d. x goes from a function of y to another function of y. So, that is the double integral Fubini's theorem says the same as integrate the variable x with the variable limits psi 1 y to psi 2 y. That depends upon y, because y is fixed. So, sum it up with respect to y. So, that is double integral. So, this probably looks more clear picture. You can see that sheet and that green thing is going to sort of move and cover up everything.