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Q2. For surface S that is parametrized by:7(0,$) ((a + bcos 0) cos &, (a + bcos 0) sino.bsin 0)where a > h > 0 and 0 < 0 < 27,0 < 0 < 20. Find...

Question

Q2. For surface S that is parametrized by:7(0,$) ((a + bcos 0) cos &, (a + bcos 0) sino.bsin 0)where a > h > 0 and 0 < 0 < 27,0 < 0 < 20. Find the surface arCa of S

Q2. For surface S that is parametrized by: 7(0,$) ((a + bcos 0) cos &, (a + bcos 0) sino.bsin 0) where a > h > 0 and 0 < 0 < 27,0 < 0 < 20. Find the surface arCa of S



Answers

$1-4=$ Identify the surface with the given vector cquation.
$$r(s, t)=\left\langle s \sin 2 t, s^{2}, s \cos 2 t\right\rangle$$

We are given a Parametric equation. Next year's along with strength, Among other parameters. You were asked Barnes the surface described by this equation. The equation is our beauty is to sign you. I Because I'm Jay plus k a friend of the must lie between zero and two We're translates easing the importance X, y and Z have except you me hold to sign you Why you equals ring beside you And does he get you ve? It's just then you have X over to where Waas Why over three squared is equal to science Where you plus science where you one So between the equation for and lips Z is constant So if it holds Econscene, we get the equation off on the lips sketch a graph of this quickly, constant z you have this level set next lifeline extraneous most to minus why it's 10 and why don't you just three at least excellent equal to zero. So we obtain summits and this is for so this is it we hold However yeah, Z is constant and we see it by the equation. Me is equality. So he is So this is also a graph. Oh, our new zien. I want you to be on. Are you Tina? And so what? This I ve tells us we'll take you to the constants, Are you not? When we get to Constance I constant I was constant J and then a variable k. So, you know, along is the axis they really only welcome zero to and so surface that is described by this creation. It's going to be a cylinder with and whipped a cool for us sections. We major axis. Oh, way three plus three is six on the why access and minor axis. Oh, feeling do. Plus two is four on the X axis and sooner has a life veering issues here too. So you and this owners oriented along the positive the axis already it's Z equals zero very long. You need to be a problem. But you say days

Were given a vector equation and were asked to identify the surface with this equation. The vector equation is R s t equals s sign of two t s squared s times Sign of we're sorry s times co sign of two t So it follows that the correspondent Parametric equations for this surface are X equals s times the sign of two t Why equals X squared and Z equals s times the co sign of two t for any point x y z On the surface, we see that X squared plus z squared by our primitive equations This is s squared sine squared of two t plus s squared cosine squared of two t Factoring out the squared, we see that we simply have signed script viscose and square, which is one. So this reduces to s squared, which we know is the same as the Parametric equation For why now there are no restrictions placed in the parameters. Therefore, the surfaces why equals X squared plus z squared. And from our lesson on quad Rick surfaces, you recognize this is a circular parable, Lloyd, with the Y axis as its axis

If that surf it's as is oriented upwards then from from latin we have the floods to be equal to face. So if we knew it's equal to that. So given that our F. Is equal to this, we want to express it in that form. So I have my next B. So this is the You see graph from 0 to bye. Then filmed several to two. So we realized that from here our this is the the region so X. Is this integral. And why is this interval? So I have my nets. Y. Said for P. You have sign why the decorated with respect to eggs. Then mine. It's cute. Is zero eggs derivative with respect to A. Y. This is with respect to eggs. This is with respect to Y. Is X. Cause why plus R. R. S. X. Y. X. Y. The mm. So we know what that is. That is X. Sign wise. So you substitute. And if you do so you have The dog being single from 0 to buy 0 to 2. You have my next Y. Eggs sane squared Y minus X. Q. Same way because why plus eggs? Why the X. The X. D. Y. So we face integrate with respect to eggs. If he integrates with respect to X. We have the Integra from serving soup. I this is going to give us my nets Y. X. A. Nine squared Y minus s cube sign. Why cause why last X. Y. 0- two. D. Y. Okay so this is going to give us so place in secrets. If we integrate you have mine it's Y. X squared sane squared Y. Divided by two. This is going to be X. To the powerful same way because why divided by four plus X squared Why divided by 2? The interval from 0 to to the Y. And we have evaluates and have servants who high have minus two Y. Sign squared y minus four. Four sign. Why? Because why plus two? Y. The way so from here you realize that we can use the trade property of Writing This as three. Right? This to be equal to one minus course squared Y divided by soon. If we do dads then we have this to be equal to Dante Iga from zero to hi I have -2 my list to y then one minus cause because to why? So two way because two way provided by soon minus two same two way blast to Y. The Y. To simplify. And if you simplify we have if I simplify I have this to be cause identical from south soup. I you have Y cause because to y minus two side to Y plus Y. The Y. Mhm. So we integrate this with respect to way and if you do that's if you eat sink needs you have one. But which is zero. Hi this but which is why? Cause two. Why do I bless I squared divided by two. So you realize that this parts is using integration. Applying integration by parts here. Buy parts and if you do so this is equal to zero. So there Um answer here is by square divided by two. So then it's in place that our flags, it's equal to pi squared Hi Square divided by two as a final answer.

All right, let's go ahead and start this question. So were given. Ah, problem where we're supposed to find the flux through S and f is given as zero I Why j minus z k. I like to put it in the vector form like that s is the region that is bounded by why is equal to x square plus eastward where y is from 0 to 1. That will be a parable. Lloyd X square plus Z squared is less than or equal to one represents a disk and height is located at why it goes to one. So I've already drawn it here. It kind of looks like a salad bowl where there's a top lid. Okay, now, to find the flux of this thing, you see that there are two surfaces, one that's facing upwards. I'll call that flux I won. And another one That's the outside the lateral A portion off the salad bowl. I'll call that flux. I too. So in order to calculate the total flux, we are going to add those two up. Okay, so let's start working on. I won so normally, what I would dio is to use the fact that if I want to calculate the flux, I find a double integral off the region s F. D s, where these guys are the vectors. But one of the theories that we know is that instead of just doing the vector calculation, we can find the dot product of F and the normal vector of that surface the S where s is a scaler now. So my goal is to basically find the normal vector, and then we're going to start with that. Okay, The normal vector off I want is very simple. It is just the plane. Why equals toe one facing upwards? So the normal vector here it is. Zero 10 So f dot n is equal to zero. Plus why, plus zero. So it's equal to why. But what we also know here is that why is equal toe one because of this situation. So this is equal to one. So it suffices just to find the double in a role of one at this region which basically is represented by the area off the circle with radius one. So we know that is simply equal to pi, and we're done for that. I want portion. Now the I two portion is a little bit more tricky because we know that the normal vector is not going to be the same everywhere. If you see it from here on the 1234 the fourth, often we can see that it's pointing towards the direction that I mentioned. But if you look at the first cock tent, you can see that the normal forces heading towards that direction Obviously those two victories air different anyhow, we should be able to find out by the parameter realization off the of the vectors. So let's work on that. I am going to say that the privatization I will use why and Fada where excess cosine theta Z assigned data why is just simply, why so the privatization of the surface is going to be call sign data. Why science data? Yeah, the partial derivative in the direction of why that's simply 010 Similarly, for that off data, you will get negative sign pita zero, cause I pita Aaron. Let's remember that to find a normal vector, we just find the cross product so it will be one times co sign minus zero, so it will be co sign. Let me make sure that part is right. Hey, that looks good. The middle part. Normally, we would do negative I times the zero times co sign zero times negative sign. But it's just zero. So I'm just gonna call it zero and the K vector it is zero minus. Let me make sure it is zero minus negative sign. So it's negative. Sign? Yeah, Negative. Negative Sinus positive science. So there you go. Got to make sure that calculation is proper. Let's take a look at F. I did a privatization, so we want everything in terms off. Why? And fada 00 Why is why negative Z is negative. Sign Beta case. So let me move some of these things over here, okay? I will do some off. The other work right here. Yeah. Okay. How far have we gone? Okay, now we want to find f dot n. That is simply the dark product off this to that. So you're going to be multiplying zero to co sign why? To zero negative scientists sign, and then we're just going to add them up, so we will simply get negative sine squared data. Okay? So in order to find the flux through I two, we are going to calculate the double in a roll off negative science square data and this part of the calculation I will let you do it. But I want to give you a side note that sine squared theta. If you want to find its integral, you can use the half angle formula or the, um, the square version of that. You will have one minus co sign off to theta divided by two. And then if you integrated, the problem should be quite straightforward. Anyhow, the thing will end up being equal to negative pie. Yeah, So, overall, what we have here is that I won has a flux of pie. I two has a flocks of negative pie, so the answer is zero. And that's one of the ways that we can do this problem.


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