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Let S be the solid bounded by the surfaces 2Vz2 +y2 and 2 _ Suppose that the density of S at (€,y,2) is equal to Set up an integral for the mass of S using cy...

Question

Let S be the solid bounded by the surfaces 2Vz2 +y2 and 2 _ Suppose that the density of S at (€,y,2) is equal to Set up an integral for the mass of S using cylindrical coordinates b) Set up an integral for the mass of S using spherical coordinates_ Find the mass of the solid S using question a) or b).

Let S be the solid bounded by the surfaces 2Vz2 +y2 and 2 _ Suppose that the density of S at (€,y,2) is equal to Set up an integral for the mass of S using cylindrical coordinates b) Set up an integral for the mass of S using spherical coordinates_ Find the mass of the solid S using question a) or b).



Answers

(a) find the spherical coordinate limits for the integral that calculates the volume of the given solid and then (b) evaluate the integral. The solid enclosed by the surface $\rho=1-\cos \phi$

So you have these region. He's, uh, some volume, you know, with dimensional space. But he's, uh, the reasoning out off the ball, given by Miss spherical coordinates is always described by, uh, roll equals true course fee. Uh, you want to compute? All right, so they really mean out of that and inside of a in the future. But he's, uh, like these hot long here and that. He's, uh so this is fears described by row is equal to two on the sea. He's be You're done. Zero. So I would like to compute the volume inside these two regions. We're civilians in between. And so for that we would like to do I need to go running in this very goal. Very cool cords. So, uh, all these region Reichel's cause fee you can be seen because you have Ah, here, roll the single from the, uh well, she d sango on road of visas. So, uh, in the plane, This plane. But he's saying the rosary since on she's young, we're gonna have this. It's fear in the row food plane where fees measured, uh, from there from the C axis. So we have the circle Ambien were date our own Vera Vera around to buy. Helping these, uh, the circle. This is here. Um, this is fear that has diameter one. And so well, for this volume, you would do as you can see, O ar fi. He's going to obtain these circle. We let he would let we go from zero. Um so from zero to there she saw the uncle that he saw by the house. So feed goes from zero through by half something we would let failure go all the way around. So there goes from cheer up to fighting is the whole turn. I'm not saying that region on the wall. The bond for role is that it has to be bigger than these because I see it has to be smaller until Onda Well, you cannot easily be let eagle from zero do my house. You don't get okay. Just points up to there way. Wilmore, get any point? Be little bay. Expect plane. So this is the correct mom range for a fee. So the thesis internal law would be all the volume will be ableto we have the bottom element. Throw square sign of fee Then the row Rovers from course fee off to. I'll see those from seer of my house Sure to by house on being era. Those from zero up to to buy. Well, some notice here. Them over the central. Nothing depends. Um well, fair. So we can do that Digital separately. You drill from 0 to 2 pi the head. I would be just about a way to meeting to buy zero. So these were people to just to buy eso We go. You can forget about the injury on, right, a factor of two pipe. So you will be equal to two pi times sold or intervals and then for Deidre we're doing roll first. So the developed world square Do you grow these people too? No, cute third's She would be that volatility. That too, of course. Offi cer that you obtain also fared from this you turn from suit by half off you roll squares up to cube to cube ar minus these You might of course, Day cube from bad. Multiply it by a sign. Orphee Uh, sign on fee on the you feel it would be just left without interval. Onda Well, here we can do a substitution. If we let ah you to be equal to course. I see, Then the over decide we have new cube. And so well, if usable to kospi the U B minor sign fill your fee, which is so almost there so that the center would turn into dangerous do by thirds. Ah, two que my guess, You cube. Ah, the U minus the u Because sign feel the fee is equal to minus the u she's have you have there. So I've been here Would, uh, have a good bones for you? Got zero on you, my house. So you're zero would be cause off zero course off zero cheaper one. And yet I have this cause of my house. These number is zero. I'm So what we can do here is we can flip being drills, and you would get a minus. So I the we make that miners go away. Only fleet being girls. So the internal from zero up to one off to cube minus you, Cube, you on? Uh, well, they drum of these, um, so, uh, internal off. That would be so. We are to buy third something times to serve our intel allies just you on the federal of you, Cube would be minus the angel of minus you. Give is minus you to the false power. You better by four on that level. Waited between sheer on one so that these with vehicle to our we're plugging in one theres do by third self tu que times One miners wanted the fourth power Very life war on that minus to give them zero My no seriously father left for so great this party cereal. And then, uh, we have these two Q b is equal to eight. You got to read minus 14 Um, these numbers physical too. The times four minus 1/4. He comes for that. He's, uh, 16 times two. 30. Jill. So say do minus one that he's 51. Yeah, I do by thirds. So, uh, December would be 32 miles 1 31 to do by 31 four. Ah, but by three on these councils, we that some of you would have 31 by over two times me or six. Ah, said these we should be. This should be the volume, huh? To that tree. One by over six

So, yeah, the following volume that is this volume, but it's bonded below Dong that you know by, uh, this surface that he's, uh, this for the role was to course fee in a spherical coordinates. Very cool on. Uh, it's probably be bonded or both about it. Bull by the car, but it can't given by sea. Simple dio spirit if x squared plus y squared. Should we look at these in the no plane? That is just Ah, row. I see we're here. These will be the C axis on the in. Our fee is measured. King down there. This is the row. So this cone would be over there giving babies line. I'm being a These regions are yourself in like, uh, like the but, uh, she's, uh it's bonded below by these curves. That is, that, uh, humbled by that is gonna be these vision. Everything. Those two under way to obtain that the three dimensions are visits to rotate. These are on feta, so that ah, three dimensional space. We were gonna have something like these. Yama here the C Axis X. Why? Yellow cone. Oh, so feel like that. I'm Dana. You have these uh, So these are cone on. Then these regions should be something like, Where are these? More so like that service. Well, that that volume and so to obtain that volume. Ah, well, the young Wolfie with the violent between these two ranges in that So these uh huh. That Kony's have being for FY equals two by over four on these range that over there is legal people's by hops. Well, so that our fee is gonna be between those two, uh, has to be viewed on fire or four smaller than by house undying, uh, because But by hopes, she's never bound. We get consign because zero. So we hit that point. Sophie's young using fiber for in my house, our fada would go all the way around. So we do the whole turn First data. So it's gonna go between CEO from Dubai on, uh, role is gonna be between zero on bond on these. He's bound is gonna be zero due course. She so that, uh, toe for a painter volume with angel. The volume drilling this vertical coordinates. So there from zero to buy Okay from by force. I'll take my house on a robe is you on? To course fee from the volume 11 t zero squared sine fee. They have B ro 50. Then there is the fare. Ah, so the reading first you can do that. His morning bear depends on federal. So the federal would be just fair. What were you looking to buy? And zero. That's gonna be too. But so, uh, the scene trolls uses that to buy nothing. That angel he's gone on Dana for well done, X. Usually you can do is in trouble. Respect. Robe. You can do it. Raw squared. Uh, the row. So, uh, you know, Rose squarely Role. That is Roque you thirds. So that evaluated meeting to cause I'm zero people. Teoh, through course fee is there Power my nose here. I'm not divided by three. So that, uh I mean thing Adriel Omagh. We have 1/3 years of to buy thirds. Then we have the center go The respect for tea by four words. Come to my house. Off to Hugh thumbs cause cute cube them Sign coffee. You see, on this point, we can make 10 years of institution because what we say that Mrs are you with you? being nickel to course fee. Ah. Then all the you will be able to minor sign, do you think? And we almost have that there. So that minors, the U musical to sign fee the fee running those terms recent development is equal to, uh oh. So in trouble, you cube minus the drill of you Cube. You when we have, ah, terms, uh, to I'm still the third. So I'm still to the fourth power times by better by through and Dana or here you would have the bonds for you by over for on you, but by hopes. So these us by house would be, uh, course off my house keys, uh, zero on this one. Because well, filed over four on, uh, before dot weaken your bladder circle. So we have Whatever. Four. If these length is one well, he's like, these are go on these prediction here with people toe course. Why? Therefore, I'm darlings there for the sign would be the same so that we have that this laying physical toe, some number a well, they call the sign will be the same. So that by the piece I want him. You have that one square has to be equal. Okay, square plus the square. So that one is He goes to a squared and then becomes You can see that 1/2 of the old way squared. So that a noticeable because whatever. For he's gonna be equal toe one over square root of two. This you didn't square it. Uh, sort of these number would be whatever is quote cool on the all this Indra would be minus new to their four You very life for on that Evaluated between zero on one over school it off to the vehicle to no minors since your fourth. My, uh, one screwed up to 3/4 power directed by four. There was 1/4. So that is this and then multiplying by to before I'm spy herd's. Um So these elites part corresponds. Used to these to this piece on the easiest people to vinyls and minus, we have a plus. I want something to square it with the fourth, but is equal to two squared. That was one for mundane minus minus with a plus. And then you'll be times that times two to the four power. I'm spy. Third's on Dena. Well, here Ah, this is the good counsel to powers are, uh, to square to square issue about four. So all those council Monday and these were illegal to buy third's, uh, which should be should be the volume All these off that region. So you should be able to go volume you. So she grew by if by thirds.

She want to compute this very goal? According it's, um yeah, the volume, but is given by, uh for C. Well, positive on DA. We'll roll. He's, uh is that all of the ball? This fear of radios? One armed, um, the it's inside. So out of these inside off. Ah, the it wasn't described by E because one plus course, he we'll see if you have, uh, there. And we do like, uh, roll the well, more over there were a few plain so, you know, we're not join Fada. Um, we're measuring she respect toe that I move so these regions is given by was gonna be, But you have to and then us you keep on going there, you're gonna go to one. I was, uh, course she, uh, as you get to the Eagles by house. Yeah. The rate is one because co sign is gonna be zero so that this vision will be like, these goes up to one Onda. Well, you would let see to continue, do like a carded, but because trying to be on oversee positive. So you would only be considering like, these vision. Since he's, um, the volume is like having this vision on retaining that around data. So, like not on the old. The region is gonna look in three dimensional space. Something like Go, do you have here Z X on? Why rob overuse one. So, like, these will mundane. Uh, we have this shape that is bounding the so it is gonna be sort of like bath shape. We're well, wait, We specialized These It's like we have this shape over the roll the plane, and then you rotate our own fatal so fair goes like our own There, Our attention. Ah, does he actually? So we're gonna have this volume we would like to compute. What is that? Volumes. We can do that in this very well. Coordinates we see. Well, one thing to note this year is that key only goes from there to there. That is the range of fears about P will be between zero on the White House. So only goes up to my house. We made this condition that she spotted, um, data goes all the way around, throws from zero toe makes the whole turn out to play Onda. Uh, well, role is between one on that number. Uh, is between one and one plus course shot the volume volume be vehicle to, um so being to go there I was from zero to buy Feagles from zero. Buy a house on the rollers from one of the one plus course fee. And then we have our volume element is your square Seidel? He they were doing euro. You seem to Max interest iffy. And then there. So one thing that we could do first these we could integrate. They're because nothing there depends on terra or residual too. You can do that. Intel of disparages Ferre. What awaited meeting to buy zero, which is gonna be able to buy. So these flu vehicle to volume is equal to our to buy pounds the central from zero to buy halves from one of the one bless cause 50 off dross square Sci fi The row, the fee. So now we can do Didn't to go there. Specter role so integrating role squarely row. That hasn't either. At the role Cube third's I'm better voted between warm plus Of course, that would be was so that is gonna be able to true by you have me third Dubai thirds. Oh, well we have betting to go left Syria from from just by over one plus coast. He, um cube minus one cube. I'm sign offi Do you feel? You know, at this 0.0, you can do a new substitution. You couldn't say that. Ah, well, let for to make things single, we could say that you is equal. Do warm bless cause thing. So we feel that do you would be good minus sign defeat. And we almost have that. He's pardoned. Respond to minus two. You there, there we have ah, sufficient to turn into the angel off you Cube. Mine is one U minus the u So minus two by thirds on dinner here for the bounds, you would have, uh, the lower bound is what ship you should be. You at zero. And what should be. You buy house. And so, uh, well, you have my house. She got one. You have by house is warm. Last course of my house. How would this party zero. So it would be you going to warm. So these numbers one and then you're zero would be you have zero is going to want plus of course, zero ableto and this number is one two people. One plus one is two to meeting through from two up to one on this. I'm doing well. She's older. Since two is bigger than one, we could flick daughter of integration on the euro minus, so that that miners will constantly be obtained by thirds atrial from one of the two. Well, you cube. Mine is one. Do you? And so, uh, will be seen through, uh, the sequel. Cool, Dubai. Thirds time times. So the internal of you cube is equal to you to the fourth. Powered by life four minus you. About two years away. Tool on one. A diesel vehicle to to buy thirds guys. Um, too good. 1/4 power. You better by four minus two within that minors. So that evaluated at once, it would be wanted them for for the way forward for minus one. Um, yeah. So you know that, uh, student forties wanted to square times Square, but at two square will cancel with these four over there, so it would be equal to two square miners to these non Berries. Uh, that is forming a student is, too. When we never decided, we have So four commissioners for wanna campesinos 4/4 to be one my nose for or four to be minus 3/4. This number and then you have a minus. So we would be to class three over four from that time stool. Why third's to buy thirds on DA. We can write these two as two of them's 4/4 plus three over for, uh, so very some need. She would be through my first of our big plus three days. Um, plus three to the 7 11 So to buy terms are 11. You got to buy three. It was four. So that these two councils up through there and you have to on the bottom. So he's not very simple to 11 by divided by three times to you doing by six. So that should be These volume should be equal to should be able to that you never by over six

To fund a massive the solid, wanting to find the region for the solids first. So the bottom is always on the excellent plea, which is very selectively a circle X squared plus y squared equals four. And for the border part, these actually inside to a rubber boys, the rubber lawyers, which is two times sex work plus y squared and the 16 minus two times X squared, plus westward these air to probe Eloise. Okay, so now we can set with integral the ami goes to away forgotten density. So the visit here is equals two square root of X squared, plus west where that's the desert Here. Now we have everything here, so access would like to the two. And wise from that could swear with four months X squared to square with a horn is exploding and see here preaching to parole Eloise does It is given bus square root of ax squared, plus westward. So from the inner to the other, we have these E b y and the ex. Okay, Now we're gonna transfer. This is willing to. It's very cool, Pop. Sorry. The polar or the cylindrical coordinates, which is way easier to evaluate to recall that are square his expert plus voice where So that's for the polar cause. Everything's pretty much in the circle. Or like, uh, we have this exporters westward, so we can very simply introducing our square here. Let's see what's going to the equation for the polar We're gonna first integrating with respect to Z and work ng lap with 0222222 And, uh, swear with four minus X squared Deposit next. Positive. And here we just, uh well, get in. Okay, now we have this. Andy. What? Easy. The ODS. So now it's the time to introducing over polar or C solution coordinates. So it's four times suited to pie, which is tthe e angle and, uh, 0 to 2, which is the radius so easily have or comes. Four months are squared, which is pretty since symmetrical. So I just did X with four ApoE four out and inside we have rt rg three that that's responds to their Christmas to the one the one e x, which is also they say the eh Thea area differential operator. So the answer for this is gonna be four times certitude pie forces or attend the theatre. So which is to itself Hi overworking team


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