 When you integrate function 1, what does it give you? The double integral over d gives you the area of the origin d, because we are taking smaller triangle, smaller pieces and then integrating. When it is a function of two variables, that is the area, when it is three variables, that gives you the volume of the solid bounded by below the curve, below the surface that is equal to f x y. That is one way of visualizing it. I think these examples are okay. I think these examples you can read. So, let me say what I want to, what is this? Examples, then again examples are there. For example, I think this will be interesting to see that example. What is that? Because let us find out the volume of the cylinder bounded by the plane z is equal to 0 and z is equal to 4 minus y. So, let us try to visualize it. How does it, the object look like? That is why I am trying to do this example. X square plus y square equal to 4, that is a cylinder. What is the axis of the cylinder? x axis because x is independent, x is not mentioned there. So, at any point x, if you want to look at what are the points in the cylinder, then they are x square plus y square less than or equal to 4. That means section, any section of the cylinder is the disc, x square plus y square less than or equal to 4. So, it is a circular cylinder. Every point in the section is a circle, bounded. Between z is equal to 0 and z is equal to 4 minus y. So, it is a cylinder. If you visualize x axis as, then z is equal to 0, that is a bottom. z is equal to 0, that is a bottom. And up to how far it goes, z is equal to 4 minus y. It is a function of y only, x is independent. So, what will be the triple integral look like? Projection is x square plus y square less than or equal to 4. So, that is a region R. So, region R is x y, say that x square plus y square less than or equal to 4. And what is the region D? The volume of the cylinder. So, what is that D? So, D is x, y and z, such that x, y belongs to R and where does z vary? z goes from z is equal to 0 and top is 4 minus y. So, this part you can write this R, if you like, you can write it as x, y, type 1 if you want to write, x goes from minus 2 to plus 2. And y goes from y square is less than or equal to 4 minus x square. So, the positive part will be square root of 4 minus x square minus less than or equal to positive part 4 minus x square. So, when you want to integrate, we will be integrating x between minus 2 to plus 2. Integrate, y goes from minus square root of 4 minus x square, 2 plus square root of 4 minus x square and z goes from 0 to 4 minus y, 1 d z d y d x. So, that will be the integral. So, that is what it looks like, d y or d x, whichever way you want to write. So, you can integrate. Let us do one more. Find the volume of the region D and close between 2 surfaces this and this. z is equal to x square plus 3 y square and z is equal to 8 minus x square minus y square. In these 2 surfaces, it is like one surface, another surface at the bottom and the region enclosed. So, how do you find out the projection on to x y plane? In the intersection of 2 surfaces, one surface, another surface intersecting. So, there will be a intersecting curve. That will give you the projection on to the x y plane. So, how do you find the intersection that curve? This is where the 2 are intersecting. So, z is same for both. So, z of one surface is equal to z of other surface. So, when you equate these 2, so you equate x square plus 3 y square is equal to 8 minus x square minus y square, you will get that surface. So, that is the curve where the 2 intersect and that is going to be the projection. Are you able to visualize? Yes. Imagine one cup like this, another cup like this intersecting somewhere. That is a solid. So, when you project it, this region will get projected on to x y plane. That is region R. And what is the solid? For every point, it goes from the lower surface to the upper surface. So, you can write down the integral very easily. So, project on to the region in the x y plane. So, and that you can write as that will be an equation in x square and y square only. So, that is a type 1. You can write it as x goes from minus 2 to 2. That curve projected y goes from 4 minus x square root 2 plus lower part to the upper part. The projection only. And the surface, so that is the, from this to this. You have only to find out which is the lower surface, which is the upper surface. Out of the given ones, which one is the lower which is the upper, because you have to go from the lower limit to the upper limit. So, you have to find which values of z gives you the lower, which values of z will give you. So, you put some values and analyze. So, that is how triple integral. So, there is computation part. So, I leave it. So, it gives you the volume of the solid. Other way around, many integrals are possible. So, you can write this as type 1, type 2 and so on. So, let us not go into all this. Again, something similar. For example, z is equal to this, z is equal to this. When you find the intersection of the two, what is the common thing? This is equal to this. So, that gives you 2x square plus y square equal to 1. So, what is that look like? It is a xy equation in x and y only. That is the curve, where the intersect. So, that means that is the projection on to the xy plane. So, that is ellipse, 2x square plus y square equal to 1. So, that is the projection. So, that ellipse, you have to draw that ellipse and see what is the major axis, minor axis and analyze. So, we should sit down and analyze these things. So, that is what it looks like. So, because that will be the common part. Next, what we want to do is what is called change of variable formula. So, that is what we have started looking at last time. Namely, in the plane, we said given a point, you can draw coordinates x and y. So, R2, which is all as I said xy such that x and y belonging to R, this notation comes because every point in the plane that is a geometric object, a plane. So, what is the way of describing it? Every point in the plane get associated with a point P with coordinates x and y and every point P with coordinates x and y will give you a point in the plane. So, the geometric object, which is a plane can be described analytically as that is a way of locating a point in the plane. And we said there is another way of locating a point in the plane. You can have a reference line O x and take a point P. How do I locate the point P? You can find out what is the distance of that. So, this is the point P. So, what is the distance O P? So, let us call this distance is equal to some R. And from this reference line, how much you have to turn to go to that line? That is the line O P. So, that is the angle theta. So, every point P can also be represented if you know what is R, what is theta. So, these are what are called polar coordinates. So, these are what are called the polar coordinates. So, if this point has got Cartesian coordinates x and y, then what are the polar coordinates? What is the relation? If I want to transform Cartesian coordinates to polar coordinates, so what is the relation? So, this is my if I want to translate that. So, this is my y and this is my x and this is theta. So, what is x is equal to? x is equal to if this distance is R. So, what is x is equal to? And what is y equal to? R cos theta and R sin theta, where R is bigger than if P is not original. So, then it is R is bigger than 0 and theta is between 0 and 2 pi. So, that is the relation. So, you can write x is equal to this, another way of writing. So, that is the relation between x and y. So, if you know theta and R, you can find x and y. If you know x and y, you can find R and theta. How do you find R? So, R square is equal to x square plus y square. R is positive. So, implies R is equal to positive square root of x square plus y square and theta is y by x, tan theta is y by x. So, you can find out theta is equal to tan inverse. So, that is the relation between x and y and R and theta. You can go from one coordinate system to another. Let us look at some more. So, this is in R 2. Let us look at R 3. What is possible? So, we have got x, y and z. By the way, this meant. So, these are polar coordinates and I should have said that for every point in the plane, you get R theta appear and every R theta gives back you a point in the plane. So, that is a one to one correspondence, geometric object and that. And to visualize this, try to visualize that at this point, if I take a circle of radius R, then all the points on this circle will have same R. Distance is same. Only theta is going to change between 0 to 2 pi. So, you can imagine in the Cartesian coordinates, you are looking at this as the corner of a rectangle. So, you can imagine the whole of R 2 as built up of rectangles. Here, you can imagine whole of R 2 as built up of circles. Every point in the plane will lie on some circle that will determine its R and how much wear on point on the circle that will give you the theta. So, visualizing R 2 as concentric circles filled up of concentric circles. So, what we want to look at is this one now. So, Cartesian coordinates, so let us look at there is a point P with components x, y and so x, y and so let us say z. So, how do you find this? Geometrically, how do you find these points? So, let us draw a perpendicular. This is my z. Projection on to the x, y plane, x is equal to 0, y is equal to 0. What is z? And having reached z, you should move either in the direction of x or in the direction of y. So, you can move in the direction of x or in the direction of y. That means, you will get these points. So, this is your y and this is your x. So, how do you reach a point? Move along. So, that gives you z and then move along the y axis and then move along the x axis. So, this is same as y. So, if you like, you can remove that point. You can remove this. So, let us say that is z. z is black, green is y. So, this is green, that is y and that is x. So, from any point, move along z axis and then move along y axis and move along x axis. So, you can see that you need three directions. You need three directions to reach a point from the point to the origin and same backwards. Given any point, you can reach by travelling along this. So, that is why this R 3 is called three dimensional. You need three dimensions and a reference point. So, these are the Cartesian coordinates. So, if these are the Cartesian coordinates, what other ways? So, in the Cartesian coordinates, geometrically what we are doing? We are looking at a box whose one side is x, other side is y, third side is z and this is the corner of that box. So, we are looking at parallel pips. One corner is at the origin, other is some other corner. That is the diagonal one and imagine the whole space being filled up with these parallel pips. So, that is the Cartesian coordinates. Another way of visualizing this would be, let us look at this point as a point on a cylinder. This is the point on a cylinder of circular cylinder of radius. What is that radius? It is at some height z. Now, how do you look at this point? How high you go on that cylinder? How high you will go on that cylinder? And on the cylinder, where will the point be? Where will be the point on the cylinder? I want to locate all the points on the cylinder. So, all the points on the cylinder are located by the height and the distance. So, z is as it is, x and y. So, if I look at this, this is the circle and it is a point x square plus y square equal to something. So, you get polar coordinates. So, x and y, polar coordinates are in theta. How do you get r in theta? Imagine this is here now. That point is lifted up or down depending on z. So, this is the radius. So, what is the radius? Let me draw it again so that you are able to visualize. So, if I have a set, that will give me this height is z. And let us say this, what color I should do? Let us say this is too big. So, that is the point. So, let us, I am just revising it again. What the coordinates? If this is the point x, y and z, if you draw the perpendicular, that will give you the z coordinate. So, I want to look at, I think this is a nice place to insert back that page. So, here is, so this is a point p with x, y and z. So, how do you find the z coordinate? You take the perpendicular here. So, call this point q. So, x, y and 0. So, this is the z coordinate. Now, what we are going to do is, on, if that point has to go on a circle, somewhere, so that will be part of the cylinder. If you look at that way, that will be a cylinder. So, to find out this point, whether it is here or here or here or here, if I take the projection of this. So, that will be this circle. Doesn't look like a nice circle, but let me draw it better probably. So, this, I should know what is the, if this is the point which is coming here, then I should know how much is that angle. See, the point is, this point p is determined by this height and how much I have to rotate, how much I have to rotate on that cylinder. Points on a cylinder are determined at what height you are and how much you are going around the cylinder. If I want to determine all the points on a cylinder, then I should know how high I am. So, that is z coordinate, up or down, that is z coordinate and how much I should rotate on the cylinder to locate it. So, to locating it, means I have to find out what is this radius. So, what is this angle is, say x axis that angle is theta. So, coordinate will be r and theta, but what is r? What is the height r? What is that radius r? So, this is z. So, this is z. So, this angle is theta. So, what are the coordinates? If this distance is r, then it is r cos theta, r sin theta, z is equal to z, polar coordinates on the circle basically. So, that gives you cylindrical coordinates. So, these are called cylindrical coordinates. So, you can locate points in the r3 by looking at points as points on concentric cylinder with the same axis. So, that is called cylindrical coordinates. One more coordinates I would like to introduce in r3 before I go to change of variable formula. So, let me write that. Probably we will revise it again next time. So, look at z. So, here is a point P. So, one way is Cartesian coordinates by looking at parallel pips and looking at the corner. The second was looking at the cylinders, concentric cylinders with expanding radii. The third is once again, let us look at this distance. So, that is OP. P is the point. So, now in the polar coordinates, we looked at a circle. But in r3, let us look it as a sphere. So, let us look at this r3 being made of concentric spheres of increasing radii. So, if I look at that, then what will be this? So, here is a sphere. So, look at this point. Look at this distance from origin that is r. So, we can imagine that point in r3 lying on a sphere of radius r. But there are so many points on the sphere. Which point we are referring to? How do I look at that point? So, to look at that point, let us look at cut by a circle of that height. Cut the sphere by a plane of height z. If you cut it by z, then if I can tell you what point it is at, on that circle it lies, that will be okay. Or another way could be, a easier way probably could be, let us just look at this is a point P. That is a z axis. Another way that could be to look at that circle would be, imagine this to be a rod, which is fixed at some angle phi with the z axis, z axis vertical one. Here is a rod which is going. How much this rod can rotate? That will give you points on the circle. So, how much this rod can rotate? That will give you points on that circle of that height z. Is it okay? So, how are the points located here? How will these will be located? By how much is the angle and what is the radius of that circle? Polar coordinates. On that circle, the easiest way of locating the points on the circle is by polar coordinates. And how do we determine the polar coordinates by finding the radius of the circle and how much you are revolving? So, if I bring it down here, so this point is z. This is r. This distance is, we have called it say rho. So, rho is the distance op is equal to rho. This is r and what is this angle? If this angle is phi, what is this angle? That is 90 minus phi. If this is phi, this also is phi. Vertically opposite angle, r angle. Parallel lines. So, what is this r equal to? In terms of phi, you can describe. What is this r? This is a right angle triangle. This is r. This is phi. So, what is r equal to rho? This is a right angle triangle. Op is a right angle triangle. So, this is 90 minus phi. This is phi. So, this is the height. So, rho sin phi. Is that okay? Height. Op divided by op is equal to sin of the angle. This is origin. So, op is r that is equal to rho sin phi. So, if this is rho sin phi, this r and this angle is theta. So, what is x coordinate? Polar coordinates r cos theta. But what is r? It is rho sin phi. So, it is rho sin phi cos theta. And what is y? That was rho sin theta. What are the polar coordinates? r cos theta r sin theta. What is r? r is sin phi sin theta and what is r? In z, what is z equal to? This is z. In terms of rho and phi, it is rho. This is z. This distance is rho and this height is z. So, it is rho cos phi. Is it okay? In the right angle triangle, that height was, this was rho. So, this was rho. This was z and this was theta and this was r. So, we get relation between x, y and z and how much distance you are. How much the rod is rotated from the z axis? That is angle phi. And how much to locate that point theta? How much you have to go on that circle? So, three things determine your position of a point. And conversely, if I give you these three things, I can locate a point. So, rho theta and phi, where rho is bigger than or equal to 0, theta is between 0 and 2 pi and phi. What is angle phi? That is between z axis. So, how much is the angle possible with z axis? 0. You can go down. So, 0 to pi. So, theta is between, phi is between 0 and pi. So, again, three things determine the points on r3. Namely, rho, how much it is away, how much you are deviating from z axis and how much you are rotating around x axis. So, these are called spherical coordinates. So, these are called spherical coordinates. So, in r3, there are three types of coordinates possible. One, Cartesian coordinates by looking at the corners of all parallel orbits. Second is by looking at a point as a point on the cylinder concentric cylinders. Or third is by looking at concentric spheres. And there is no wonder that these spherical coordinates are very useful when you want to describe the earth. Any point on the earth, you want to describe. Earth is a fixed radius. So, rho is fixed. Imagine a sphere for fixed radius. So, what are other things that will determine how much you are away and how much you are rotating. So, these are what are the longitudes and the latitudes of any point on the surface of the earth which determine your location on the surface. So, if you go to Google and you want to find your location, it will tell you in terms of phi, longitudes and latitudes, how much you are away from and the base, how much you are rotating. So, these are very useful in modern navigation, modern location of points on this sphere. So, and we will see how they are useful in our mathematical thing also.