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42 0ousEvaluate the surtace Integrit Js Teads where $ ts cone wlth pNrametrlc equation ucon(w). usin(w), 0s"<1 0s 3n72Show all work on your paper Hold your...

Question

42 0ousEvaluate the surtace Integrit Js Teads where $ ts cone wlth pNrametrlc equation ucon(w). usin(w), 0s"<1 0s 3n72Show all work on your paper Hold your paper up for the camera: In the space provided enter the page number for your workEdit Vicw Insert Format Tools Table12ptParagraph

42 0ous Evaluate the surtace Integrit Js Teads where $ ts cone wlth pNrametrlc equation ucon(w). usin(w), 0s"<1 0s 3n72 Show all work on your paper Hold your paper up for the camera: In the space provided enter the page number for your work Edit Vicw Insert Format Tools Table 12pt Paragraph



Answers

$42-46$ Find a vector function that represents the curve of
intersection of the two surfaces.

The cone $z=\sqrt{x^{2}+y^{2}}$ and the plane $z=1+y$

Oh, spiritual coordinates. The value tripling to grow over eat. That's comes right time. See Eli between the spirits really close to and rode forth and about the cone buying fortune. Hi, over three. No, we're already given E in terms of spherical coordinates. All we really have to do. He's just evaluates with triple in the roll over e the x Times y Tennessee. This is not liberated. Integral. Um, well, we said we're above become five. Chris Pyatt three. So interpreting from 50 I was tired lately, integral from fading equals zero to time in the world from road equals to two very close four. When you want to write a function in the spherical coordinates, this is a road towns signed by consign data times wrote times sign five times Sign a leader and protons Coastline of five Tanzi Differential, which is very squared. Signed five dear. Oh, data Fine. Even for Dini's theorem, you can like to see the product of integral so substantial growth easier high of the three of signed Cute. By who signed five. The five Alexander grow from 0 to 2 pi um, from sine theta sign data total comes down to grow from 2 to 4. Oh, go to the fifth you will taking into derivatives. We did when the U substitution in my head. This is 1/4 times the sign of do that. So five before from zero to pi over three lines during another institution in our head this is one half sine squared data 0 to 2 pi and then time. 16 36 24 And he spent particular for our second factor here. Rural into this whole into grow believe zero.

Yes, We want to integrate this. Ah, triple integral of X y z Devi over E which is our region between really close to and Rick was four and above are Cohn Pi Phi Z equals Pi over three. Now we're gonna go ahead and evaluate this so we can go ahead and convert everything to spherical coordinates. So we know exes. Roco sign data went by. Why is Roe sign far sign Data sent by and Z is Roco side by. So then overall, our x y z become so, um X Y z is equal to row cube Um sine squared If I co signed by sign there. Ah co sign data. It is a mess. And then we have DV, which is roast where it's signed by the row the by your data. So then we'll go ahead and drop all of this into an integral with our bounds. We know. Um, go 0 to 2 pi in data. Ah, we're above pirate three summer go from zero to part three in row and fire me and we'll go between two and four in row. So then we have ah wrote to the Fifth Sign Que Phi sign the co sign. Oh, sorry. I forgot another co signed by co signed by sign data Geoscience data the roadie for, uh, you follow I d data. So it is a jumble mess of intervals. So we're gonna go ahead and try to solve it is best we can, um and we know that our first part, it would just be wrote this 6/6 at two. And four, which gives us four to the sixth, minus two to the sex over to which is simply just, uh, 2016. We don't have our inside its 2016. So have that. Bring it to the outside. Then we'll have these inter goals us. And we have signed cubes if I co signed by science data science data. If I data and it's a simple U sub since sign Cisco satisfies attribute of a sign signed cube of just sci fi, we can you know we can do it. You seven or head. Where we have sign is you. And since you cubed is just for over one this integral on the inside. This is just signed to the fourth five before from zero to pirate three. Uh, he data, and we know that. Ah, Sign of Pirate Three is his screw. 3/2 and then four. Think fourth power gives us 9/8. Sorry. Not over 16. So then we have, um, 2016 times 9/16 integral from 0 to 2. Pi of I forgot the rest of this article. So we're just over signed Data Co Santana, And then we'll have just that part of the end. Um, data, I miss another use up where we have signing co science, and we have science court. Over to then we have 2016 times 9/16 of sine squared over two data from 0 to 2 pi and notice that no matter what, uh, he puts two pi or zero, you get zero for sign so our overall integral actually just equals zero.

We have the land situation one upon f physical Toby one upon the I plus one upon the oh, where d a is the distance between lens and the phylum and D O is the distance between lens and the object. We need to simplify this equation such that the question contain no complex Tom. So by simplifying it and by taking Elsa off right inside, this become one upon f is equal to B d oh, plus d I divided by d o multiple i d. I know we can relight it as value off if this comes out. Toby after taking the sip Local of the question If this comes out, Toby Dio multiply greedy. I divided Way the o plus D I. Now we need to find the focal and if it is, given that the distance between object and the lens is X centimeter and the distance between lens and the phylum is two x plus one sentimental. Now we're putting the value of the O and G I. In Equation one F is equal to be X multiplied with two x plus one divided by X plus two x plus one. Now, by simplifying it vocal and comes out to be to access to power two plus X divided by three x plus one. So this is a required focal int.

We're getting a surface and grass peas computer with pretty graphing software to grab this surface services the equation X with minus six y four y squared minus Z equals zero. To graph this freshly write the equation as a function of X and y So you have that Z is equal to X squared minus six X plus four y squared. And I'm going to graph this surface using Dez Moses Parametric surface graph for and here is a good view of the surface. So I have the opacity of the surface turned down so you can see the inside as well. You see, this is as our equation indicates, this is a elliptic Carrabba Lloyd, which at the center that is not the origin. The center is actually looks like it approximately or zero negative 10, perhaps negative nine


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