Hi there. Prior to that, in space curved surface area accounts We have developed methods to. And a repeat here If we make a compilation of six have found that the calculation of ways. Basic and general approach With this we show d s mockery of the surface in space, is the area projected onto the plane xy If the unit of surface area with the z axis perpendicular vector the interior of the unit vector in the direction perpendicular by multiplying this formula is obtained. In many problems directly n are available and that account The formula ever developed a need for would be able to do without. But the subsequent formulas are improved formula. It is basic, given the overall result According to the type of function by calculating able to develop. An outdoor function of the first kind when given, The function described surface, n is required here when he calculated is that of s curve in space surface in the x-y plane integral in terms of the projection of d We found that it was. Similarly, a closed surface If defined with the function here again by x, y and z as function partial derivatives can be calculated. In the third type of surface with a vector function u and v are two parameters, but from the parametric representation d s is the projection in space denominated in the area of ??the x-y plane We have seen that such a relationship. Here again, the length j Jacobian emerges. The two binary j x i of Jacobins, the row i and milk threw the remaining two binary matrix determinant. This is a Jacobian. je l'first year again when we account After I remove the line and the second column the remaining two binary determinants j gives the determinant of the matrix y. Similarly, j k. j s get it here The length of the vectors. These are more fully removing the saw. So far, all we know in the plane by integrals, but in the plane of a two-storey When we want to find the area of this terms was always going ahead. When a curved surface, but this terms that reflect curvature we choose the function representation by type is calculated in this way. A new method, the surface again the parametric representation and you have to calculate in terms directly. This is a curvilinear coordinates u and v are. V held constant when x i.e. only one of the vector u becomes a function of the parameters. This is kind of a vector function We are aware that the line in space. Similarly ui sabitleyin de x v is a function Another line of family forms. And in this case curvature surfaces in space d u d v not only their Cartesian coordinates would be if you had a j s. In this way an extra term for these comes to the calculation of the integral. A special type of a surface In the area of ??surface of revolution We have also developed the formula for the calculation. z in the vertical plane as a function of r If to such a representation, geometrical surface of revolution about the z axis of this line a surface obtained by rotating. Now their various We will see examples. While our sample following a well we will do the calculations in detail. d s is the appropriate solution here d s is the surface area is calculated as the integral. But usually this kind x y plane of the surface from the center of gravity length is also important. According to a z axis, The second moment is also important. They also will do. When it calculates the first torque with z d s h divided by slamming it, this surface along the z-axis gives the center of gravity. Now let's move on to concrete examples. The surface of a hemisphere We want to examine and the scope thereof six different We want to calculate the road. First, why hemisphere? Because of this sphere projection by the time we get to the x-y plane, a circle in the upper hemisphere, will be projected onto a circle, The projection of the southern half Also the projection sphere, thus take twice I need to find a full sphere. Therefore hemispherically We are working on the come on top of the projection lest occurs twice. Multiplying the results obtained by two We all find the area of ??the sphere. We know that already. Do you know that since time immemorial. Since the dawn of high school as well. The surface area of a sphere We know that four pi squared. When a radius. Now let's start the general notation. We know that such a formula. A half sphere equation representation of the sphere diameter x squared plus y squared plus z is equal to the square of a square. In this formula, n we need. We need to know this n. n is the length of this surface gradient We know that division. But the steep gradient vector gives a length not necessarily. A length that obtained here to provide The gradient vector we divide longitudinally. Very easy gradient, derivatives with respect to x two x's. derivative with respect to y two years. two z z derivatives. Because every time the other variables for doing the hard tasks. When you calculate the length of this four x squared plus y squared four four z be the square root of the square. Now comes a simplification immediately. Because we know that, sphere from the equation, x squared plus y squared plus z squared, is equal to a square. Therefore, here we put a square Remove the square root of a. Also here are a four. This is the time when the square root of two and would simplify the above two. Therefore n x y z position becomes a division of the vector. Unit size that we can control. You are calculating the length of x squared plus y z squared plus the square root of the square will be. He therefore squared equation, out of here. So this vector we k in the z direction, ie with we take the inner product of the unit vector, k is zero is zero. Therefore, zero times x plus zero times the sum of z to y once. Here the denominator remains the only z. There are already a denominator. He already will not be affected by this process. Thus, on the surface of the sphere d s infinite There are small areas in this way. d a d x d y n k'yl internal multiplied by z, we have split. This is of course because it is in the denominator of this mold a share of our time goes in the denominator. Such an integration. Should we write it in Cartesian coordinates z rather than a squared minus x squared We will write minus y squared. As you can imagine this sphere, x axis of the hemisphere, x y axis on the x y plane of will occur in the projection of a circle. For this, the appropriate Cartesian coordinates not coordinate circular coordinates. R d r y for him d.times.d d theta are writing. x squared plus y squared de We know the r squared. Then the circular coordinates d s in this way is it more simple. Now there is the issue of this integral account. Usually, such that the square root We are trying to get rid of the square root of time. Therefore, a squared minus Let's say u r squared. d r we need. Minus two times r d r d u will. Already here we have r d r. Therefore, our work will go quite smoothly. See here for lack of theta, integral removable suit. Reset the integration of two theta pi a full circle because we are wandering. the relevant parts of r'yl d r r r squared minus a squared. You can now do the integral We find the value on theta. Here there was an a. Here comes a two-pin. Two pin a. This term r squared minus a squared minus As first force and if we define r squared minus we call the square, this u minus one-half second would force. If r d r d u would have a negative divided by two times. This integral immediately the integration can be done. Limit one should pay attention to the course. A square is happening is zero. While RA is happening is zero. This integration is very simple, defined integration. Get minus one-half of the integral Do you know a We are increasing exponentially force. When you add a trailing minus a half a split in two and one-half we divide into two. Divided by a slash of course, goes to the two share. Here we had the minus sign. This dilemma was to simplify each other. There are minus pi. Now it at zero when calculating a square when he calculated at zero, here comes two of because it is the square root of u. Because it comes with a minus sign This also improves the negative. We find two pi squared. Because it results in a known All of the surface area of the sphere According to a squared four pi is half will be two pi squared. As you can see no formulas without memorizing only the perpendicular vector knowing that from the gradient able to take account of the convenience. Here again, a common meeting integral type, have a square root. When the square root generally see the transformation of our business. In the second method implicit function Let the representation. The open function, but also have representation As you can see with the function off sphere equation easier. Gradient are account. These f x, f y, f z. We take it for putting in the formula instead. That's our formula, d s is the formula:f x's square, the square of f y, f z squared. You can see them when we put Nothing like the previous situation we encounter. Gene x squared plus y squared We are also a plus z squared. Comes out of the square root We find here a. Take the square root of four will be two. This dilemma also in the denominator by two sadeleÅip found previously divide z d times to come. Of course, the next integral is the same as the previous one. Because d s the same. And this circular coordinates We did turning. We're doing the same thing here. If we make representations with clear functions, See there the formula were as follows:x is a plus for There squared plus the square of f y. You have the square root. Now get some more of this integral As challenging but not so difficult. See derivative chain rule, Every time again how we see that useful. A squared minus x squared minus Do u say y square, that one half of the second force, one-half of its derivatives will fall forward. over one-half of a will raise will be divided by two minus one. D u d * will be also divided. He divides d u d x is a minus two Give the equation for x, that these partial derivatives can be customized. fy is done in exactly the same way. x instead of y. Our formula is a plus for us x squared plus y squared had fun. F x squared plus x squared denominator for a Because the same common y'yl We can now combine the denominator. A squared minus x squared minus y squared. y frames are also here. Now when you bring them to a common denominator See squared minus x squared minus y where a There is a square. There is unredeemable plus x squared plus y squared. x and y take each other, remains only a squared denominator. There is nothing in the denominator is already changing. Was the common denominator. You take the square root of d a'yl We find d s gets hit. Here are the previous two is here s is the same expression is reached. Therefore integral to again, you do not need anymore. Because I've done it integral. Altogether this integration. Also found here are merely d s. Now the parametric representation brings a new dimension. A new type of solution method. We have to account for it, this j. j u and v by a partial was the vector product of the derivatives. Now of course, the surface of the sphere I need to know the parametric equations. a radius of the sphere such a parametric representation. If you say u fi, theta, if you say you have. And v is the request here more concretely, The following is happening and theta functions. So fi fi varies along the curve We're going through the meridians. Tete geographical terms the way again If we use the latitude you're going through. Our previous coordinates Judging see here, When replacing the plug Open're going through. When replacing the theta We're going out latitude and one way of them an area is achieved. But of course every time they You do not need to go back and do. The benefit of this formula. Now this by x the plug If we take the derivative is easy. From sine cosine will be. Others are the same. Here again it will be the cosine of the sinus. From here the minus sine cosine will be. V, i.e. by theta When we received the derivative, the See where a sinus fi common to both. Minus sine cosine come from. From sine cosine will be. In the last term, the last component, In our theta no. Therefore, the derivative with respect to theta is zero. Vector multiplication means the first line will write i j k. The second line will write the first vector. The third line will write the second vector and we will calculate the determinant. Since there are i j k course will be a vector. See here have a partner. We take this outside When there is no content. Here, too, has an a. We also take out When we create a square. This i the first to find components We are temporarily closing line. Vertical are closing. Here zero plus sine-squared fi cosine theta involved. Similarly, the term j'l are account. Similarly, the term K in composition are account. Here in this vector, we find the vector j. This gene of the vector length When you calculate what You will see that much simplification. Already it Jacobins We saw the same sample. Because we calculate the Jacobian from the also nothing more. Now where e, j's easier to find a frame. Then take the square root. it is a four squared. See it on the square had four sinus. There cosine square. Here are four sinuses. There cosine square. Sine, see here are the sine-squared. Therefore, the common sine-squared as going out. Inside the sinus remains bi sine-squared than four due to its square. But this is a time cosine in a time frame, sinus multiplied to the square, meeting. As you can see here a bi occurs. See here that when there is a once again a very interesting structure b. KosÃ¼nÃ¼s square sine-squared fi fi is happening. He is also a giving. So the inside of the brackets completely. These are not coincidences. Characterized in that the global koordinatar, nature. Therefore important is happening already. As you can see how much was simplified. When we take the square root of it Of course, we find a squared sine fi. Here when we calculate d s, divided by the squared sine f j z. z j from here we know what it was. This is a squared sine cosine fi fi. The following are the more simple sine see here. I sadeleÅtirel one of a square. Keep a a a a below above. Because I see a time where the cosine fi in the z coordinates are gone global. X of the first component, the second component y z the third component. This simple geometry We found with the projection. If you do not remember the formula is given. Here the denominator means that a occurs. In the denominator of the z ouÅuy. So this is already one, two and three the same that we find our approach. Thus be of integral will be the same as it should be. Now here you have an interesting deÄiÅikik. The following respects u and v coordinates directly in terms possible to calculate the integral. See the previous page j s were calculated. j s, has a squared sine fi'y. You do not need to recalculate. By doing this these accounts can be found, of course. Integral j h V. Di Di was once the general formula. Although means V d, and d d theta d f can say. As you can see again the integral variables are separated. Because it does not function at all in theta. Therefore teta'l sections are separated. going out for a frame to be fixed. Here on d theta integral b will give us two p. Backward integration is to plug the sinus fi d. Let us also note the limits. We are working half the sphere. Theta hemisphere from the Arctic, sorry f is equal to the equator are coming from scratch. There, a ninety-degree fold would have. So it's going pi divided by two. The following is the sine integral. This is also very easy to EUR minus cosine fi. The above limit, below the limit value. See here for two pi squared occurred. This is the first integral and at the beginning of coefficients. This cosine of the above fi'y value zero in pi divided by two. Remember Flat trigonometric circle. Angle is zero at the North Pole. Cosine of the angle is zero. Fine, f is equal to zero when you put the cosine becomes zero. But there is a minus sign. Therefore, these two minus one another taking this integral remains zero. Here are the two pi squared. It should already gene We find the same result as. This approach is completely different. In all of ÃbÃ¼rki curvilinear hemispherical surface so the x and y We were getting to the plane of projection and views. A circle was happening there. He's on the circle in the plane used to calculate integrals. Here it directly fi out coordinates and theta in space on the surface We calculate the integral. And a supremely simple integral output. No conversion is needed. This is because the global spherical coordinates surface of the natural is the fact that coordinates. So this curvilinear coordinates such benefits also need to remember. Surfaces of revolution, we know that. But if you receive a half-circle, I'm sorry if you receive a quarter of a circle, Such a half-axis in a quarter circle. When you rotate it here the surface of the hemisphere is formed. Thus, the function z is equal here A squared minus x squared minus y squared, from the equation of the sphere. But for x squared plus y squared is r squared, z is equal to the function f r We find here. According to this variant should run. Here you force a split second he approaching easier to calculate. Here bi with a minus sign one-half times together. Insiders derivative is minus two. Therefore twos and in the denominator goes here minus a split second force is going. One plus for the square of the base We find it again when you get in we have encountered before and completely integrals we encounter. This gives a known result. But see here on theta integral surface of revolution is beginning to and this formula was taken from the inside lies in the effect of theta, theta contribution. For him it is a surface of revolution Because a convenient time automatically that an integral You can reserve in formula. Contents of this formula that includes theta, contains. Now I want to give some homework. Follow the same steps, how some of the course immediately You can see that, but I would advise you to do. weight, based on x-y plane We want to find the center. Our finding that the first moment z'yl have obtained multiplying. In this area divided by the area We have calculated that pi squared. It's the center of gravity. You to account for this integration I want it to six different way. Bi in the spherical surface of this technology, naturally in the study of many events something that may be encountered. A machine element of a global consider physical, substantially or an electromagnetic space on a sphere consider the distribution or A heat conduction through a sphere of with the distribution of the heat flow passing here You can deal with such problems. And here it The second moment would need. The second moment of x squared plus y squared. r squared x squared plus y squared anyway. This also accounts for this I'm waiting for you to do. As results given in You can check your answers. Yet there's an assignment. A previous version of this a little more special. We are one of the hemisphere We found the surface area. Hemisphere rather than a cone of here ALSA or something like the truth to take away the arc of a circle Even if you turn about the z axis not a half-dome, which hemisphere but not a dome for zero clear, making an angle of, for zero angle with the z-axis less than a half-dome that you can find. More generally becomes a problem. Here is a side view. Eliminate them if you receive a section, these cone to two, the manufacturer is going on. A dome that circle arc of a circle to be obtained. What is the difference from a previous problem? There are very few differences. In spherical coordinates I want you to do this. Because you can not do in the other coordinates. You can, perhaps, but supremely You can meet with integral harder. It is not very difficult but more of a integrals must be tall. Hemisphere for zero angle pi divided into two parts was coming up. However, one for zero When we come to terms, Open for zero for zero comes up. sphere but has a yarÄ±Ã§apÄ±y If we say is, x and y are in the plane. Circular coordinates are in the distance. This is the sine of a fi who is possessed is reset. Current distance. This lower rim of the dome ring This is not the semi-dome r would be too small. Reset is for sinus. These guidance as The information given to you. While this area accounts the limits of integration from scratch You will receive fi to zero. In the previous problem from scratch had received up to two pi divided. Similarly z'yl hit d s, the integral from zero again f you will get zero. rar hit square again from scratch You will receive fi to zero. Therefore, previous equiv problem completely, equivalent in these results we can say f to zero if pi divided by two, See for pi divided by two is zero if we do Our hemisphere comes dome. Efe, the cosine of 90degrees is zero for the cosine fi becomes zero and back to zero be two pi squared. E also know that half of the surface area of the sphere. Similarly wherein kosÃ¼nÃ¼s for zero fall. The cosine reset for fall. There are such a control. There is a supply of the compound of formula. Gene homework. A cone area and moments I want to find. Where b is a cone surface of revolution. But it's the way all kinds of I want you to account. Because that's our goal a little not very random surfaces, integral, not too messy, and simple integration with significantly on surfaces improve our skills. This cone, cone, since elementary school, the maybe some of you might have seen. If you cut to the middle, If you turn in a plane such The surface of the cone ring segment means. Height h, a is the radius of the base The length of time the generator side square is the square root of a squared plus h. For this, the problem here is While you will need the following. Need cone equation. This cone of the equation x squared plus y squared minus the a and h are when given a squared minus h z square or a square times circular coordinates If you see translations where r squared x squared plus y squared where z has to be square. Everywhere that squares Take the square root of z is equal to the right where the interests of the equation. Divided by the height h of the slope of this line. See consistent formula here As we come to this conclusion. Here in the software For convenience, plus h squared divided by a square, If the square root of alpha surface and torque and weight center involved in this way. One more example. Let's take a paraboloid surface. Dimensions along the its length to come out I have a one-half to divide. Here are the x squared plus y squared. z equals three divided DOUBBLE roots, in the times of surface with a plane that Let's cut the surface of the paraboloid. As the situation is going. There such a Paraboloid. This even as a surface of revolution a Paraboloid obtained. If you say x squared plus y squared r square There are a function of z is equal to r squared. This paraboloid surface different I want to find ways. N find a single thing to be aware of. E n very easy to find. Which kind of representation of functions If you receive be easy. Moments already given here. They see it as the moment If you do not want, with physics If you do not want to take care of, see them as an integral. ÃÃ§neml find the d s. The steps in the same sphere We have solved a very detailed here, If you follow the answer to this You can also find. Now here I gave the details. Once you get back before hesaplasak as a surface, z is a function of r as x squared plus y squared had. If it is not square, we find this formula. Upon receiving derivatives here by r, As you can see the dilemma goes. is a divided leaves. Here we put them This formula is a plus f is the square root of the square base d r'yl integrated'll hit. Why had received three roots? Because the root of three is taken A scratch is going to. The root cause of these three. Here are the steps that we make, As we encounter the integrals. In this integral the usual approach again We are trying to get rid of the square root. Gene is a plus, squared divided If a square is also here When the two are di account d is a square divided involved. Putting a square here, r d r have found work in this way we are. This is obvious. So it's that simple an integral comes. This integration is not difficult to do. Of course we have to look at boundaries. u is zero when r is one. where A is R is A, See u is from here, but there are squares, result here, we find the integral. General impressions of the road Shall we go now, just Akinde but it is the latest method of surfaces of revolution We take advantage of the paraboloid surface. Thus, single storey integral've done. The two-storey integral to the overall We need n my representation. E z given us. z f we find the gradient immediately. Steep gradient vector of f Recalling that We are adding a third component. He does a negative one. To remove the third dimension here we find by dividing the length n. c squared plus y squared de We find that relates to a z'yl. Of all these transactions, if we once k'yl will take it to the inner product. So it will fall to x and y. K is zero because a zero components. Get the absolute value of this negative mark will remain a going. There is nothing to do in the denominator. Place them as a denominator comes. KarekÃ¶kl ??phrase comes to share. On the other hand x squared plus y square We know that the r squared. D in circular coordinates x, d y r d r d theta know that. Therefore it is completely just the came to see that the integral. Because d s are the same. Integration is performed in the same way. with display off function get close to the right by z is equal to zero we can. Where x, y, z gradient by take. x, y, z here see from the gradient by There are two x's coming but at two in the denominator. x divided by a. in the denominator in the coming one to two years by two is more simple, y divides a. Take the derivative with respect to z by a minus. Always a gradient vector and surface perpendicular vector. Is a length at the moment we do not know it. In order to do work for him f of x squared plus y squared, z squared plus f ie x squared plus y squared, A common denominator of all Although we have taken a square is coming. the square root of a square of gold as a It comes with the same integral comes again, r squared x squared plus y squared is happening. Let's open with representation that's to say the previous problem. With parametric representation Let's make this more a new dimension to the count brings to solving the problem. In this method, we need this gene. j parametric equation of this surface also as vectors r and theta terms are writing. See here x r cosine theta, y r sine theta to say, ie at z r squared x squared plus y squared the from both the cosine square that is based on a square sine square We know that a divided by two. So we found this on our own, but we find easily a parametric representation, depending on r and theta a vector function. Yet according to the variables, We take the derivative with respect to the parameters. See here, here, from the derivative with respect to r only cosine theta, only sine theta, here two in two r A Sadeleer r divides. Theta derivative work again in derivative of cosine theta minus sine theta. Derivative of sine cosine theta-theta, but this The last component is not in the theta zero for a derivative thereof. theta is obviously remains In the first two terms in derivatives. These two gene vector Place the determinant, we i, j, k in the first row, type x more exactly, the second line of i, j, the following second line of x r k Put this determinant when he calculated again as usual for component i We take the first column of the first row. See, here is minus zero divided by the square of the cosine theta times. ji'li a time when we look at We start with a minus sign. jima are taking the columns. Cosine times zero, zero. There are more negative plus a negative is happening here. But from the beginning of a negative There still remains negative again. Gene r squared times the sine of theta is going to split. And when we come to K in composition components see here is the cosine squared theta times minus, minus, plus one more r squared times the sine of theta. Sine cosine of the square A square of the sum of where r is staying only. This gene vector. Here we find the length. As you can see the length of divided by the square there is a time, There cosine, There is a square one is doing their sinuses. There is a third term squared. She obviously from the first term There is a squared divided by four. How consistent size You can also see that. is here a uzunluks have a length squared. Wherein the length of the fourth power means divided by the square of the length The frame length is happening again. They are providing the interior. You take the square root of the As you can see when There is a square one here is above all a, There is a square where a run, If we take her out of the square root of r is happening. Under the square root of this will remain square. I stayed here r the r squared Because we get out. Here was the one. He squared divided by r squared. We know the formula. Curved surface area of this jar of s j x, j z section. We find that z r j. So d s, j s here r divided we found a split r. As you can see is takes but here too there is a will, You can run if you want the gÃ¶tÃ¼rttÃ¼r. The same thing. In the square root of a plus square is a square remains. There is a well here. d r times r, d theta. See a denominator is there a run here There denominator here is another one there. Hence r still remains open. M s so that the same first, in the second and third methods d s have achieved the same result. Integral is the same. Now here's parametric If we work with a representation of x, but it is function of u and v, If you have x's representation here at theta We wrote see as a function of r and theta. So the process here When derivatives by r, theta derivatives, we find them. j s, but also have already found by following the same steps We find multiplication of vectors. Once the j s r d s d, d theta. where j s. d r d theta in this circular coordinates it is the surface of revolution , as a natural result of the results we have achieved. Natural coordinates because b r and theta rotating surface. I'm giving a paper. Still the same problem, the same geometry, This time the same function z d s is the integral over time. Times Square is a well run s is the integral over. To do this, a lot to do There is nothing more than a. We found for d s. Will take to put the square of this integral There is nothing other than to calculate. Something extremely simple. Please enter the six different ways We're saying, but already a In previous problems at all d We have found that s the same. Therefore repeatedly There's no need to do. Just out of this integral We see that go. Results given in. The results we have achieved gives the ability to control. Have such a duty. This time again, a paraboloid but the downward facing surface paraboloid surface of the dome A paraboloid surface. Here is a number 15. When we give our zero x squared plus y squared. Will be the square root of 15times. Why 15? Because it simpler to calculate 15When we get involved. Because a close to 16. Accounts by little try you'll see a lot simpler. The previous problem with exactly the same steps and with very similar accounts that something will be a In previous paraboloid of a upturned were cut in the horizontal plane. Here is the downward pointing paraboloid We're still taking a nap in the horizontal plane. This horizontal plane slightly a special horizontal. z is zero horizontal plane. Z is equal to the previous one was a constant. Calculates exactly the same path is achieved. Completely analogous to the calculation of the integral. A simple example:see the hyperbolic cosine. Hyperbolic cosine function is a parabola but it looks a little different parabola. For example, a form that occurs in nature. Please take a chain assay at both ends under its own weight as it is hyperbolic cosine function is happening. Technology application in our vine up and down our bridge cables is the hyperbolic cosine function. A full one hundred percent of the ends free Since there are an open-chain on your feet, but a little buckling hyperbolic cosine function gives a very good approach. Why hyperbolic cosine? Because integral exceptionally here comes simple. This paraboloid of the same or A similar geometry. E it e problem Follow the same steps you'll see the d s always going to be the same. But for cosine integrals taken Supremely will be easier. The direct conversion without integral will come. Now bi en surface We are coming to the integral. TR surfaces such as our bi Or a wheel Or, as the car's steering a The shape of the car, such as bicycle tires receive, but the center of a circle Since the z axis within a b a b Do dÃ¶ndÃ¼rÃ¼yos around the longitudinal axis z. If we look in three dimensions, hoop receive Are you turning around the z-axis. If we look at it from above the work The diameter of a circle here this apartment Are you walking my way. It's the middle wheel When you cut the shape. Or bi tire middle The view when you cut it. Of course, on this There are such semi-circle. You want to find the surface. Now it will pass quickly. Do you also look a little bit because it If you ask me, but have located the how well we do it How do we find the equation? The easiest to get here with parametric shape representation. Because when you look at that kind of circle In any such bi center of the circle equation. Therefore, the x coordinate Such a circle See the point on the b'yl we come up here. Circle of radius a. If we say here in terms of f, the projection would be a time when we received the cosine fi. b got here, too. Distance from the center axis. b plus a times the cosine fi. When we look to y, y's times it's a sinus fi. But this is not x and y, r z plane. We return because we take it that's when the car tires, bicycle tire, a piece of equipment, We wrap coil bi electric utility We find a way. So where x is on a mission. y z makes the task. z at a time sine fi. So this taurine parametric surface representation here are welded. If we go one step further we need to find x, We need to find y, We need to find the x position z Let x and y components of the vector z. Now this was r. If we take the stand that r cosine teta'yl completely in circular coordinates We find theta x. Teta'yl sine of the same r If we multiply y we find. We have already found z directly. Place them by the we find the vector x. I said that the easiest way parametric representation. Here, in terms of its x y z If you try to write what how much difficulty can imagine. Z can be found for sinuses means whereby You can find it in relation to r'yl. Currently, there are cosine terms You can solve something goes. But this parametric representation See how immaculately turned out. Theta and plug attached. We're taking the derivative with respect Fi. We're taking the derivative with respect to theta. Take the product of these two We find this gene vector. here i j k'yl it first In the second row vector, the third line of the second vector As we put. This again when he calculated by multiplying simplification studies interesting is happening here. You can find it a bit long, but When you get the length thereof, When you calculate the rather involved in a simple phrase. Although the apartments with b is zero on the We immediately see that return. Sorry to those of sphere, because it When passing through the center of b We have rotate around the axis. How to find the surface thereof theta and by the need of integration based on the plug. As you can see with some functions fi teta'l separated part of the function. Teta'l part of the function is very simple. A. Why one? This is because for a rotating surface. Once d theta. See here also have an a. Of the two, two pi future. If you receive this, the integral There is something very interesting. The following is the cosine of the integral will be zero. Because the cosine of a full period If you calculate the plug plus area minus the area will take. If you do not want to compliment it it is the sine integral. Sine the value is zero, the two pie. The value zero is zero. So the integral cosine drops. b times two pi remains. And this tire, wheel Or in this way the surface of the wheel You can see that. This second moment I want to find. So is it hitting square Integral to do b. d s's this particular job but a little reinforcement hands If you make yourself think. Consider both the such circles, but tr shapes such as used in the art. Many nature as a model system You may not find the same in nature but b can be closer as a model. With this circular coordinates I say found. He is a surface of revolution we found for our bi previous r of the function of this structure function is found. d s is what we know. This integration is performed. Gene circular coordinates I want to find moments of torque. The first moment I did not want to find. Because the weight of such a tori center of the coordinate axes It's obviously in the center. Simple symmetry in this way with thoughts we are treated as a serious example. Now we're standing here today. Because then in space We will now integral to. But this time the volume of surface integrals. The volume integral, volume object volume requires three dimensions. So we will do triple integrals. Bi goodbye until further your opinion.