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Which of the following are parameterizations for the paraboloid z = X + y2 ? (select multipla) K(r , 0) WK rcose,rsine , r2 Kr,0) =K rcose,rsinb _ 76,Y) =< Ky- F...

Question

Which of the following are parameterizations for the paraboloid z = X + y2 ? (select multipla) K(r , 0) WK rcose,rsine , r2 Kr,0) =K rcose,rsinb _ 76,Y) =< Ky- Flr") =<r

Which of the following are parameterizations for the paraboloid z = X + y2 ? (select multipla) K(r , 0) WK rcose,rsine , r2 Kr,0) =K rcose,rsinb _ 76,Y) =< Ky- Flr") =<r



Answers

Identify the quadric surface as an ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, elliptic cone, elliptic paraboloid, or hyperbolic paraboloid, or hyperbolic paraboloid by matching the equation with one of the forms given in Table $11.7 .1 .$ State the values of $a, b,$ and $c$ in each case. $$ \begin{array}{ll}{\text { (a) } z=\frac{x^{2}}{4}+\frac{y^{2}}{9}} & {\text { (b) } z=\frac{y^{2}}{25}-x^{2}} \\ {\text { (c) } x^{2}+y^{2}-z^{2}=16} & {\text { (d) } x^{2}+y^{2}-z^{2}=0} \\ {\text { (e) } 4 z=x^{2}+4 y^{2}} & {\text { (f) } z^{2}-x^{2}-y^{2}=1}\end{array} $$

One Today we are going to solve a problem. Number seven your function of sequence Breathe their day plus four x day plus to my kids. So closing the global sea if dark the are because double integral over Sigma Bill cross if dirt and the yes C is defined us ex commonly it's a square blocks away square calls for Zodiac was zero. Closing doesn't oversee. If not, they are is equal to you. Can Google Overall, they will cross f not the the the artists return us ex commonly minus two less than a record legs less than a record Normally less than or equal to square it off four minus X squared So d is given by de off X ways that which is the goto the plus x squared plus no, it's square minus four They're Thadzi equals So let's say plus what because Okay, so they'll cross f is it going toe? I dialects trees It the Yeah, don't right, forex. Okay, Does it do what? Which comes to be the white because three D plus four kids so lying. Illegal foreign level of success. Yeah, a little of the seat f dot They are because minus two integral minus square Root off four minus X squared, too. Four minus X squared the way, plus three day plus four K in north to accept plus two y j. Let's kick Hindu Delight DX, which is equal to minus two toe in the minus square. Root off four minus X squared. Two for minus X square. Thought it. It's plus six way plus four in do the way. The eggs, which is equal to zero total by 0 to 2, are in do for our cost data plus six are scientific data plus four in the the digital so which comes to be four from zero toe by Because data data, the roto our square. The year plus six Hindu 0 to 2 pi scientific data the data into the zero to to our square the plus four into toe by my zero and 0 to 2. Our deal it comes to be a bite can do our square by two 0 to 2, which is 16 by. That's the no for question. Thank you

This problem were given six equations for surfaces and we want to identify what kind of quadratic surfaces they are. So here I've written the six different forums of quadratic surfaces and kind of with general coefficients like a BNC. And so we'll start with the first one. So we're kind of familiarizing ourselves with the different quadratic surfaces. So we have six X squared plus three y squared plus four z squared is equal to 12. So first, what we want to dio is when we have a second order terms in X, Y and Z, we want to normalize the right hand side. So we'll divide through by 12. We have one is equal to X squared over two plus y squared over four plus Z squared over three. And so this looks like an ellipse oId with a equal to the square to to be equal to two and C equal to square to three. Next up, we have y squared minus X squared minus Z is equal to zero. So noticed that Z is a first order term. So we're gonna reorganize this with Z is the first element, and we're also gonna make its coefficient one sold, but we'll multiply through by negative one. So we have Z plus X squared minus y squared is equal to zero. This looks like a hyperbolic crab Lloyd with a equal toe one and be equal to one. And next problem. We have nine x squared plus y squared minus nine. Z squared is equal to nine. So again we have a second order terms on the left hand side and a constant on the right hand side. So let's divide through by that constant to normalize, it will have one is equal. Thio X squared plus y squared minus. So sorry y squared over nine minus c squared. And since there's one negative term on the left hand side, this is gonna be a hyper high purple Lloyd of one sheet. So then we have a is equal toe. One b is equal to three and C is equal to one. Next, we have four x squared plus y squared minus four. Z squared is equal to negative four. So we're gonna normalize the left hand side started the right hand side, and so we have one is equal to so negative X squared minus weiss Word over four plus z squared. We have to negative terms. Therefore, this is a hyper Boyd of two sheets with a equal to one. Be equal to two and C equal to one. Next we have to Z minus X squared minus y squared. Sorry. So minus X squared minus four. Why squared is equal to zero. So first, what we want to do is we want to normalize the coefficient. Is he still have Z minus X squared over two minus two. Y squared is equal to zero. We have a linear term and to negative terms. So this is gonna be and elliptical Prabal Lloyd with a equal to square tissue and be equal to square of one half. Or we could write that as, um or to to over two, if you like. Next, we have 12 z squared minus three x squared is equal toe four y squared. So here we have all second order terms. However, we have no constant on the right hand side. So this is an elliptical cone. We'll move everything to the left hand side and then we want to normalize the coefficient, etc. So we have Z squared. So we're gonna be dividing through by 12. So is the squared minus X squared over four minus y squared over three is equal to zero. So then a is equal to two and B is equal to score three and that completes this problem.

For giving an integral. We were asked to evaluate Simple and the role of X plus y plus C off the region K, where he is the solid in the first, often that lies under the tabloids. Ecause four minus X squared minus Weisberg While this parable Lloyd Z equals four minus X squared minus y squared, this intersect the X Y plane, which is the plane Z Po zero in a circle X squared plus y squared equals four or influential coordinates. This is R squared equals four, and then a Zara's positive. This implies articles, too, and so and so into a coordinates. Our region E If you set of triples our data Z, it's the fatal eyes. Well, because when the first often realize between zero empire but to are is going to lie between zero and two and Z will lie between zero The X Y plane and between the Paraiba Lloyd four minus X squared minus y squared. Reaching rectangular, rectangular or mixed. Switched to yeah, political coordinates is four minus R squared. And so the triple Integral um, experts y plus z over the region E. This is the iterated integral, which is integral from 80 to pi over two and drop Marco 02 in the Balkans equals zero for minus R squared of a function in terms of cylindrical coordinates. So this is our cosine theta plus our sign data plus Z 20 differential, Which for cylindrical coordinates, this is our times Easy BRD data within the anti derivative with respect to Z we get and they grow from zero to pi over to you go from 0 to 2 one back going out on our hold Arkan's ours are squared co sign data plus sign data eyes mm plus one half times are times Z squared from Z equals zero To see those four minus r squared DRD data with the anti derivative Sorry. Evaluating integral from zero to pi over two integral from zero to and this is or are squared minus art of the fourth Times Co sign the A plus sign data plus one half times are times or minus r squared, squared DRD theater and taking the anti derivative with respect to our this is integral from zero to pi Over two of here this is four thirds are cute minus 1/5 part of the fifth times Co. Sign data plus sign data and then this next term. Do the use institution in your head We get, see one half times negative one half times one third This is negative 1 12 times the inter function or minus r squared to the new power three from r equals 02 Data and evaluating you get integral from zero to pi over two and then plugging in. This is 64 15th times the cosine of data plus the sign of data and then plus and 12 times for cute, which is 16 thirds deep. Data taking anti derivative with respective data. This is 64 15th times, man man, today of your fear is going to be signed data minus cosign data. A 16 3rd data data equals zero to pi over two. Evaluating you get 64 over 15 times one minus zero plus 60 16 3rd times pi over two minus 64 15th times zero minus one minus 63rd time. Zero because zero and we simplifies to two times 64 15th 1 28 15th plus a thirds pi

Do you want to find the volume between the salads? Ease equal to 18 minus X squared minus three y squared. And the solid Z is equal to X squared minus y squared. Now, we don't really have to draw a picture just yet. Um, we can just look at the fact that if you put in plugging 00 for X and Y Ah, you see that, um, this is called us when he won t to so at the origin. So at two u zero Ah, you will on its credit and see to So we can assume that for our region that we find our volumes. Um, that will have rz one function is that this creator there is he to function. So first we want to do is find a regional are And how we can do that is by finding the intersection between these two, these two ah functions we have Does he want to see what is he to? Or 18 Minus X squared. Minus y squared are three mice word we write that better. Three y squared Zico Two X squared, minus y squared. Now, with a little bit of, um, just a nebulae. Xinmin, get this. 18 is equal to two X squared plus two y squared. Leaving us with following X squared plus y squared is equal to nine. Corresponds to the following polar equation. R is equal to three. So since we have a circle with radius three is a region are we know that our just described by ours lesson We go to three and create within very close zero, and they does between zero and two pi. So now we just have to integrate z do this double integral over our You want money? Easy to D A. Now, if we make this polar will notice we'll have two things here. Go have X Will have, um Well, go ahead and actually just do everything after we were put this in the next in line. So we'll end up with They want to grow over our of, um 18 minus. So we end up with minus two x squared minus two y squared D A. Converting this to polar coordinates leaves us with the winner over our of We have 18 minus two R squared. Artie, Artie, Data. Now, if we go ahead and consolidate all this, we end up with informs your pie interval from 0 to 3 of 18 R minus two r Cubed you are data and now we do our normal separations end up with following since they're not dependent on each other and we'll end up with two pi outfront times, um, nine R squared minus, um 13 end up with 1/2 are to the fourth in 2 to 3. And this gives us two pi times. So at 00 sweet and a progestin, the three parts we have nine times three squared or nine times nine is 81 and 1/2 times, um, arts to fourth, which is 81/2. Um, leaving this middle part becomes 81/2. Or this this part here is 81/2. So what is stuffed with 81/2 times two pi or 81 pie? As our final answer to this problem


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