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Let D be the region in space where y 2 0 0 < ? < 3 and 2 2 12 +y?_ Hints: D is half_ cone balanced on its point at (0,0,0) , with height 3 and radius 3. The a...

Question

Let D be the region in space where y 2 0 0 < ? < 3 and 2 2 12 +y?_ Hints: D is half_ cone balanced on its point at (0,0,0) , with height 3 and radius 3. The angle that the curved sides of D make with the positive z-axis is T/4.points) Write an iterated integral for the volume of D using rectangular ordinates_ with the innermost integral with respect to and the second integral with respect to y. Do not evaluate the integral.points) Write an iterated integral for the volume of D using cylind

Let D be the region in space where y 2 0 0 < ? < 3 and 2 2 12 +y?_ Hints: D is half_ cone balanced on its point at (0,0,0) , with height 3 and radius 3. The angle that the curved sides of D make with the positive z-axis is T/4. points) Write an iterated integral for the volume of D using rectangular ordinates_ with the innermost integral with respect to and the second integral with respect to y. Do not evaluate the integral. points) Write an iterated integral for the volume of D using cylindrical cOor- dinates_ Do not evaluate the integral. points) Write an iterated integral for the volume of D using spherical coordi- nates_ Do not evaluate the integral:



Answers

Let $D$ be the region in the first octant that is bounded below by
the cone $\phi=\pi / 4$ and above by the sphere $\rho=3 .$ Express the
volume of $D$ as an iterated triple integral in (a) cylindrical and
(b) spherical coordinates. Then (c) find $V .$

So have the region region He in the first opened the first opened under. He's bounded by this fear. Um, the inside this fear Ah, he It's fear of radios Ruggles three on Uh, yeah. Bake on Described by coun Do you buy? She equals fight over four Eso is gonna be the helper region so we have the angle here by a floor that uses a cone. It's good to see because 5/4 you have the school Very nice being We cut because either considered a region inside of the screw reduce three. So something like that, we're he's, uh, zero physical toe tree on the school is given by fi sequel to, um, by over four. Well, she would like to sit up in this very gold gold in it. What is the volume? So as we see here, our uncle he is between zero from that angle there. That is fiber for feel. Leveraging Syria by over four. Uh, well will go from zero up to three World's goes from zero pill three and then you won't do consider just the region. One day first talked him, you would do You'd have to cut like that sort of the only obtained this portion. Well, being for that portion, we would be through a street. Thiago Vieira. So what we have you here, X Some. Why? That actually is there. You see, Data is going from that. Like that. Actually sexy obtained by said it goes you on the Y Axis is obtained by saying that later Siegel do by house. So there will be between zero ne buy a house. And so the volume would be well fair. Goes from zero up to I have so that that would be Vera. Mom, feed goes from zero. Okay, by were four. You I mean and rose from sear up to three. You roll. I'm Dana. Was this region the first off them on the volume element? Is there a square sign off? So this is the volume in this variable. Do you want it to consider the volume in cylindrical cylindrical? What we have for federal, We have, uh, the san Restriction, because this it up in this very believes the same fit us you since cylindrical coordinates. So this will be the range more fair in that case. And then you have the restriction between are. And, uh, so you have, uh we don't need to concede there these portion. And so, uh, well, here you would need to have here. Are you have here, Um, we have here, See, on the that. He says there's are so as we can. Not this Ah, for a Z between sear on this point, it's called that see, Not, um very These are is gonna go from zero. A deadline that is even by c is equal to Are these the beaches? The angle g equals two by over four. But that angle is not is gone is not in cylindrical coordinates. So we have to express it like that. Are you able to see? So that is, uh, for C between zero. I'm see, not the restriction for our is that our goes from zero, uh, from zero up to to see and then between. Ah, well, that point there should be three. Because I have a radius off our sphere. I'm being for C line between See, not I'm free is gonna be given by these. Oh, ladies, curve. So that our we'll go from zero of the boundary of that curve. The curve is this curve here is, uh, c square you described through C square blocks are square. Ableto three squared is nine. So that our our goes since it is positive you go up to nine square root off mine minus c squared mine c squared. Um, so that would be making God range on Dana. All the only thing that we need to All right, find out is what what he's ever seen. Not what is that. See? Not not. And so we can find that by saying that, uh, well, on that point So she's thesis angles by over for these distances, three so seem not should be able to three. That is that these US times sign off by over Well, off by over four for that. Well, that angle, by or for on the sign off by over four. He's, uh, one over. Square it off to. So we're be the BBC. Not no. So a decent Drago are fair. Go from sear up to by house that he's the data well mean are or a scene what was gonna go need to separate it into two pieces. So from, uh, from Syria, Tool three squared off to the city for you see? And then we should have another interval. Ah, from three squared off to. So we should have another deal from by saying that. Um, see those, uh, from three. Very square it off to of the three. These so that these 200 girls cover the whole range. We'll see. You could see blue go from zero up to three on then in these fish first, integral Well, r release are was from Syria to see So that would be the are on the volume element is just are she would be that. And then over the central there goes from ah, sirrah off the square root of nine miners see squared So that is for BR being before doing well in the SAR. So these should be volume Onda, uh was you guys he It is easier to build this integral. So let's do that Interval. So, for it is William. You know Indra from Cyril up to by house wolf the, uh, Feagles from zero to buy over four. No, the sea rovers from zero off the three year Oh. Then they be helpful. We'll square sign, throw square sign or feel so Uh, well, we can compute the interval. Respectful, uh, data. We feel our first because nothing here depends on fear. So getting teary that separately the center got would be go to just fair. I evaluated meeting by house zero. That's gonna give us by house. So you have my house terms. All these are very intervals from zero by four from 0 to 3. Rose squared. Sign off. We grow Onda. Um, all the drilling for no squared the role. Is it really possible to row Cube third's? I said that between three on Syria with people, too. Three Q third's my no cereal. So these councils want powers. He would have three squared. There's gonna be equal to nine. So he's internal goals away when we get on nine. Teoh vehicle, The Central Legal Tau nine by, Huh? Mind my house. I'm being being a girl from zero off to buy over four off. Ah, sign. Oh, Do you feel I'm being trouble? Sign of fee. Hey, Intel of sign off we Is it really possible to minors? Cross key? So you do literally not meeting zero by over for that That is equal to minors. Course five before minus cause of zero on the clothes off by over four people to one over a squared off to because that's here is one that one can also be written as a school off to or a skirt off to so that you will confine us one minus x squared off to no holder. Square it off. Do it Seems we have a minus that makes you to be able to score a total minus one. Oh, very. Square it off. Um, sort of these will be photo the volume of people too. Do you have that? So nine by halves there's around there being terms that you know very well that times screwed off pool miners one over a squared off to So they should be the volume When, uh, well, perhaps we could simplify by multiply allies. Quote of to I'm tough on them bottom. And so if you do that, you get the physical to ah, you know, to the it's on mine by And so tu minus squared off to over, um, growth of two times that. So two squared people before so nine by as to square it off to Well, we're for So you get Ah, So the volume musical toe these Where do you prefer that which are the same

You have the falling volume, you know, a sphere of radius for X squared plus y squared plus c square. All that you say share racist, too. So this is fear. Well, um, so this is fear. And then you would like to find set up the angels to find a volume well over Virginia's inside of these. His funeral rings through on DA We'll see. The people who should not see is positive. So the other part I'm also inside inside of the cylinder given by X squared plus y squared Siegel, do just more on a warm ceiling. There they're spiral to the C axis. So, like these, she would like to set up dangerous in Ah, it's cylindrical coordinates to find his volume. So inside of that sphere, I mean, cylinder, miss idled. The suit in there on the side of this is here for senior year. It's here. So, um, for the volume, you know, cylindrical coordinates. Um, you want to Sarah the integral or order of integration as the c. R. It's fairly well the volume element and see in Tripoli's are and so you diploma tries see, in terms of off our so as you can see. But you can not This bizarre in this recording it he x the Why, uh, she would go from zero to there. Tiu have Ah, something like these, if you have. Ah, are I see? Oh, you have equation that are These parties are squared. So are square lost. He square has to be smaller than four. So that, um we have this condition so that all these helper part would be since sees positive before possible with that, but it can be written us. See most of these mourner down the square root of four minus r squared my by moving. These are there. So, uh, you have that condition. So that c was between zero up to the top that is given by God are floor minus x squared. Um, Argos, uh, from zero upto one. Because we have this condition better. Why are square has to be more than warm inside about together. So our is from 0 to 1. I'm being all fair ago's ah, all the way around. So the whole turn. So for me, that would be from zero Dubai. So that is if we wanted to compete disorder. Now, Um do you want to make, uh, you want to do? So you want to do? Ah. Now for border integration in the form off you are. You see There, you know you have to think of. Ah, are will be. This is the volume element. So I was in here. Are he Turns up, see? So well, I was gonna go from zero up to one up to this point over there where it is given by this condition. So our bryce, uh, but this year, that one from zero up to one my he's been up to that point that point, he's, uh were you would have are square. Plus, he square Well, that's in these curves. That is that at any point before, but to you have r equals one. So you have the question, but one plus c square single before. Remember, C Square Siegel do room for minus worm so that Cisco is gonna be squared off. C is gonna be squared off free. So for decent ago, You do well, So our goes from we from zero off to one r r c. He said I'm being role see Goes from zero to squirt of three on the fate all the way around from zero. Opted to buy, but then beyond of Ah, screw the three should be See in that case would be from square it for free up to two. Jeez, the point here that he saw to my house. Yeah. Zero Leicester Square before. And then, uh, for that point, um, from the from that point, we would be in the curve that are is gonna be for my nose squared, squared that Qatar. Where are Dems? From there to They're four from zero before minus C squared. What? Um, but he's so for Are you are on the ground Is for you on the theory goes Taro's from zero to play. So that is, uh, you got two. That interval now You wanted to do the integral. There's off. Uh, you see your but we have the same volume element on being fair. Goes from zero to buy on. The see is the same A sooner Well, on this Cindy's into go more See, uh, booze. Who she saw Do wolf see goes from ce goes from zero up to Well, now this equation c squared. Plus are the square cigarette before, so that C this mom is upper bond is given by screw the four minus r squared. So see, Goes from zero up to screw it off for minus R squared undying. Well, our bulls from Syria off the one because we're on these conditions that are square. Must have wanted at one. So that is a bone for the Are so these party you hear? Indeed. Drills? Yeah. That one on Dean, no.

What do you want to go? You are being degree. You see, It means you call coordinates called AIDS. The following. We want to find sort of being two girls to find the volume off this region are between. So these are inside of wall. Well, reviews, uh, do centered my religion. So we have soames fear so that all these looks like the, uh, So it is, uh, you sell these more, but it is also move the plane Sequels. You. So that, uh yeah, it is about the plane Siegel zero on these ball is described. This is furious. X squared, plus y squared. Plus the square is it? Although square, she's four armed. It also lies inside the ceiling there, over these one, um, really through. Well, going along the the sea direction. So? So these ceiling, they're here. He's, uh x square was why square one. So it is, um, inside the cylinder. Um uh, well, between between those two being those two. So as you can see here. Ah, well, you need to go. Coordinates are there, Uh, on c. So all these region is, uh, he's It's symmetrical, respectable rotations. Carol, because, um, yeah, you seem that triggers. We go. You don't. So So we want to set up the angels to compute the volume off these regions. Body require. So first I want to do the c r BC We'll see if you can see there. Ah, there you saw O r. See So well are goes from zero off 21 because our is equal to the square root. Thanks. Square. That's why it's weird. And they know where are you see is limited to like, No, these curves which is, uh, more screwed for my square So that well, there is a factory in derision off are you are seagulls from zero off the screw the floor Minor, sir. Squared from being r r Goes from zero of the one on the Sierra Goes from zero to buy. So this will be we want to set up these order I will give you there. That should be the volume are now, um you know, because you're wonderful in order. You are, you see? Said, uh um So now we have to see are as a force General C Also as you can. Not this year. Um huh. The the ladies are It was all the way of Dorne for C for C, um is more than these number or here so that no more order is when his number is when see, you see Walter R. We'll see you today, uh, for good or for my nose One, because there is one. So that happened with the physical to school for free. So for C smaller than its code of three, these iron goes from zero are Groening's from zero 21 So that it will be the singer girl will be are the are are from zero off the one Condon's E v t zero squared off free. I'm being they see are the terra. Okay, Rose with the zero onto pie. Plus we'll know the point where where our dependency is below this region so that you need to say, Well, R c has to be bigger than squared off e I'm not Well, it's gonna be, you know, too, because But he says, quote the four and then, uh, all right, well, that gaze will be with zero on, uh, squared off, uh, four minus C squared. So these distorted bones for Oh, Marcie, I'm for our help. But if he is zero on this group, the four minus c squared When they get out. Well, the same video Kogan you see then with respect, Professor, that our ingenuity to buy. So that is, Ah, if you want to compute this volume, that order will get that on dinner. You want to do the order? He said, uh, you see, you are She's, uh that does not dependency on on our these, uh, order of integration. Well, is the same as in these one. So you have balls. He depends on our because there is the last of arrival. So you have the Jacoby in. I mean, we'll figure there goes from zero to to buy. I'm gonna be where I will not come. She's, uh see. But you is, You squirt the four minus our square zero, Miss Cooper or my hosts are squared on. Uh, are are those from sear up to one before? So that is that is that

We're getting a great y squared Z squared over this um shape. So above the cone Row equals pi over 3? So it's a really fat cone. Um because power for three is 60°. So from here to there is 60° And below this fear row equals one. That's the green thing. Okay so row is starting here at zero and going up to the sphere. So row will be going from 0-1. Oops I meant to erase the mess I made there. Okay row equals zero to row equals one fee. Always starts here at the Z. Axis. So this is equal zero and then it's going to stop when it gets to fee equals pi over three because that's the answer to comb. So that's what's keeping it inside the cone this fee equals zero. Duffy Equals five or 3. And then tha tha well it's the whole cone. So data has to go all the way around to get the whole cone. So data 0 to 2 pi. Yeah. Okay so now I'm just gonna leave those hang in there. Let's see why squared. So why is rho sine fee sine theta. And that's squared. And then Z squared will be row co sign fee squared. And then devi is rho squared sign fee dear O. D. V. D. Theta. All right so here we have a row squared. A row squared in a row squared. So we have road to the 6th and we have a sign fee squared and a coastline fee squared. And another sign feast. So we have sign cube fee cosine squared fee. And then we have a sine squared data. Okay that's only one of those dear. O. D. V. D. Theme. Okay do you have the morning? Okay so we can go ahead and integrate row to the 6th dear. Oh and we'll get Road to the 7th over seven from 0 to one. Thank you go some squared sine squared. Do you feed the rope defeated data? Okay so that gives us 1/7. So let's put that out here. Yeah. All right now here I have a odd number of signs and an even number of co signs. So I'm going to take one of these signs and split it away from the other two. So I have sine squared fee, coastlines squared fee sign fee D. V. And then we still have the sine squared theta D. Theta. Yeah. Okay. The reason I did that because I want this to be D'you so I'm letting you be the co sign. And so d'you is minus the sign. So I need a minus sign in here which will put a minus sign out here and I need to replace the sine square. So now I have line is 17 integral integral 1- Cosine squared coastline squared minus sine squared minus sign. Sorry the fee. And this one was the was 0 to Pi over three. Okay if you equal zero I'm sorry if uh V. Equals zero then you equals the cosine of zero which is one If he equals pi over three Then you equals the cosine of pi over three. Uh huh. I have three square 321 Which goes 1/2. So this turns into 1- one half. One minus U. Squared U. Squared D. You. It's one half that's U. Squared minus U. To the fourth. D. You. Okay so you cubed over three minus you to the 5/5 from 1 to 1 half. So 1/8 over three -132/5. Mhm -1 3rd -1 5th. All right. I'm gonna work on that over here. I got one 24th minus one third minus 160th plus one fifth. So this needs an eight and an eight. So -7/24. And then this needs 32 minus plus 31/1 60. I can't Mhm. I'm just gonna do this minus seven times 1 60 plus 31 times 24 Over 24 times 1 60. And then I'm going to reduce instead of trying to sit here trying to think of what the common denominator is. 160 times seven so minus 11 20 plus 31 times 24 7 44. Why nestle. Okay, so now I have 376 Over 24 times 1 60. So I'm going to divide by 218, 8 Over 12 times 1 60 94. six times 1 60 47 over three times 1 60. I'm not liking that very much minus one. Seventh times 47/3 times 1 60. Hope that was negative. And now I have to integrate the sine squared of theta. D. Theta. So 47/7 times three times 1 60 I have to put in an identity one minus the co sign of two theta over to this data. So 47/7 times three times 1 60 times two Integral of one with respect to data is data-. Can we need a two in here to make the you and another one half. So one half signed your data We're going 0- two pi. So 47 times seven times three times 1 60 times two times two pi. Mhm. Yeah seven time oops seven times 3 times 1 60 16. 33 60. Dude. Yeah okay let's see if I can. Yeah yeah yeah okay so I paused the video I'm not sure what that looks like on your side. And I did I recalculated and I got minus 47 of 4 80 so I'm not sure where this seven came from. So we got 47 minus. Uh huh. Then somehow it turned oh yeah it's this 7:00 47/7 times for 80 times pi. That's what I get for the answer. It's kind of gross but I can't find anything else wrong


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