Hi there. Now to calculate volumes received the work we do. So we're looking at infinitesimal volumes. We saw Cartesian, We saw cylinder coordinates, We have seen the global coordinates. We've seen it before a specific body type. Of surface of revolution We have seen accounts. Now, here in the same way rotary surfaces. But the house is full of it as well We want to find in the volume. Topics are as follows:z is equal to f r Whether such a function. Considering this is in the z plane, Suppose a curve is as follows: from A to B r, z also get up to d from c. Now, this is a curve in the z plane. But if we think of space, r, theta f r and z is the z plane and when a function of theta, the absence of theta function. We know that this is a special surface. Because it is independent from theta think of such a surface in space Take your time alrÄ±sa which theta r z will come as we see them. This equivalent to on the opposite Coming:take these spinning z Turning the axis. Here we see in three dimensions. we rotate around the z axis When we get such a surface. But it also is full for We believe because we will find the volume. Before this I only We were concerned with the surface. Here in the plane of any theta, theta equals b x y plane of a fixed If we think is true, but in space, it's right Built on a plane. So take a knife to the surface Suppose we cut each theta for the same curve means to occur. Here is the independence of theta. Here we show this form. Now, we've seen it before. V d v r d r d theta d z times. We know that the cylinder coordinates of Theta is independent of the volume no theta in theta integral this place does not appear teta'l separable portions. Thus, here comes just two pins, p go to zero when the two theta. But there are two options in front of us. Now there is where d, d z there. We can do on our first integral or Before we can do the integral over r. This tells us two different case reveals. See, before we take our integrated on Let's write it as a three-storey bi ago integral, theta from zero to two p. we received on our first integral z is equal to the value of this constant c where z is held constant in the plane r c value on the curve f r values ??up to the situation. You're writing. d z integral from c to f r. From then on the r integration will take. There are times d r. Here from A to B is also going up. If we consider here is in space radius goes to a radius b. Tete on the easy integration. Because no one here theta does not appear anywhere. His coming two pi. d z on easy integration. Because of the integral're getting one here. Z is one of the integral. This will put our d f r ago. We will put our d c, will raise. So such a one integral to the story's going. Because the surface is special because theta is independent of the three-storey Instead of a single storey integral integral account can reduce the problem. Similarly, We previously held constant z r As we change the constant z ago r can also change the hold. I.e., wherein the integral run away before order integrally on the z We can do the integral over. Two theta pi will b on again already. Our function f r z is equal to our away.The Considering that the inverse function r in terms of z will function. r is denominated z function, say j. So, r is the integral of the square and j z limits will be between. Why? Because here we keep z constant. This is also a value of r is coming up. E the r value with a j z are giving z height. z height is not here If the values ??are different here will turn reflect that. E the r integral is easy. r squared divided by two. Two, two in the denominator, wherein the two There are two from a pier. Staying back with a pin. Then r is square instead of the j we will put the j frame. Where r squared minus for the following limits, A square is going on here. d z. As you can see here a special surface Because structure, In the absence of theta three a single storey storey integral integration can download. Already these formulas univariate function when reviewing the able to achieve independently. But here is what I like about you I hope you will also enjoy things of this formula from triple integrals suspended in a special state. There is also a sense of each of them. See our f r minus minus c so hard, like height. Here we find the height. There is here. Surrounded by two pins which, when multiplied, Once such a radius r means to have returns. This gives a surface. Gives a cylinder surface. Height at radius r f r c is negative. You can also bring a thickness because it is the delta. Because here there is also d. d r is also from this part of the delta. Thus was obtained a thin shell, a Are you going to have to give something like dice. A cylindrical membrane. Collecting these membranes, i.e. an onion Think of it as the surface of each bulb, wipe kestik consists of a membrane, you collect them The total volume of time you will find In this way, you see. ÃbÃ¼rkÃ¼ If the situation is somewhat different. z j pi times. See where j z e, the radius. j z even the radius from here. This radius squared pi When you multiply it by, As you can see here going out of the circle area. Remove this pi times the square means that where a formed around it by means to remove the area of ??the orifice. Pi squared minus a squared times j Such a ring area. You are multiplied by the delta z. That is a fine size on z. This is similar to a stamp, such as backgammon. Annealed in the middle of washers Get rid of depression, z such that a ring thickness of the delta. Here are obtained by collecting them we are going volumes. By summing these discs, the thin opposed to the collection of shells coming. This three-dimensional overall The meaning of the first integral z is done on or before opposed to on coming. That's as geometrical this thin infinitesimal volume shell cylinder, cylinder ring something like a stove pipe volume. If you're here as thin as a backgammon stamp shows the volume of a disc. They were able to slightly genellet. Z in the previous example a I was going from fixed to variable. But a curve in the lower limit can be provided with. If z is equal to the lower limit thereof is g height at any point From that height f r g to remove the r is obtained. This gives the height of the cylinder. Around its z this is returned when Is worth two pi r buluyos wall. When you give the delta is Bi thickness Do ediyos that's getting thin membrane. Similarly, E, z of constant values ??for a From curve b in the previous example to r is a constant value of b as We varÄ±yo variable value. We found based on height, According to z values ??of these variables have varÄ±yo. But in the beginning it was not fixed can be selected as the variable. It also means that e, j squared minus h j squared minus a squared square but when The two curves function of z may be between the volume. Wherein the values ??of r going between the two. So as a full're getting thin. When you turn it just work I'm talking about as full of plunder because this is a hollow square pi g h're taking here. This, in that the first term outer radius obtained Or also gives space and delta z'yl We find the volume gets hit in d z'yl. Moments that they might be interesting. There also are given them. The center of gravity of such a surface obtained by rotating to be the center of gravity integrals on one of our bi, To find the center of gravity in the z direction. around the z axis the moment of inertia, To find the second moment it is should be multiplied by the square of the genre over there. These are provided in this manner. Z'li it is very difficult to translate. It is very difficult to translate the r'li. For this reason, these two accounts This known system also torque makes it easy to find. We are now up to our operations where bi compile. In Cartesian coordinates did you know the situation already. In cylinder coordinates We learned supremely easy. But a little bit of their geometry We endeavor to understand. Because of good practice b To make the essence, I need to know where it came from with the principle. This annular area d x d y We stand where d z'yl. Likewise Cartesian coordinates in the plane bi rectangle area d.times.d We stood still z'yl y d. The global situation is slightly different but After quite a few accounts supremely elegant, simple terms bi emerges. Here is a bottom gun, a top for a limited r'yl rotational surface with a thin membrane that account. Is pi times height times two. BI gets hit in the delta r'yl The thickness of the membrane is happening. Here, the backgammon 've done to stamp simulation. PU gives times radius squared area b. It would remove the empty space in the middle. So this area is going to ring. When it hit the d z'yl We find infinitely small volume. In all these integral the following questions come to mind. So far, of course, just endless small to digest, volumes were found. In various embodiments hereinafter After starting from the Cartesian circular and a combination of cylinder coordinate We will see examples can be used. Both of these concepts will be reinforced We will see both methods of calculation. These are in many applications, professional issues, technology, the nature of the subject Choose from the integration will occur. So do a bit artificial, To do so the mere account refrain from the sample. In all these integral As two basic questions 's team which coordinates the use? Which option do? The answer is simple. If you object to oluÅuyos of plane You will use Cartesian. A cylindrical symmetry Or if there is a global If you set the coordinate geometry Your choice will be compared to him. BI's going well now. There are three variables, see here. Which is the integral over do? In which to do? This in turn limits the answer from the simplest in which he let veriyos. Now I'm gonna stand here today but now I want to show. Examples here anymore, I'll start with the problem resolved. Some of you I'll leave you to do. Cartesian ago I'll start from the coordinate. Then circular cylinders and spheres alternatively coordinates can be used together We will see examples. Sometimes someone much more easily, sometimes the other is that it is easier to be seen. In this way, both of these concepts We will be reinforced as well as the calculations will learn. Bi goodbye until more opinions. I hope you will digest them thoroughly. We have found a simple delta. But it is useful to understand a little about them. Here also two methods. A simple geometry, bi de jack calculating jakobiyan. Goodbye.