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Conven the intcgrall f5" courdinaler And E alunicVvedd Immnecunourcvlindric eLet E bc lhe colc {(5,X2:v7m4} Usc cylindrical coordinales find thc volute, VoEb) ...

Question

Conven the intcgrall f5" courdinaler And E alunicVvedd Immnecunourcvlindric eLet E bc lhe colc {(5,X2:v7m4} Usc cylindrical coordinales find thc volute, VoEb) Find#Jlz &V, the z-coordinate of the centroid of E:Use spherical coordinates find the volume of the solid between the sphere r" +y+2' = 9 nnd the cone rty

Conven the intcgrall f5" courdinaler And E alunic Vvedd Immnecunour cvlindric e Let E bc lhe colc {(5,X2:v7m4} Usc cylindrical coordinales find thc volute, VoE b) Find #Jlz &V, the z-coordinate of the centroid of E: Use spherical coordinates find the volume of the solid between the sphere r" +y+2' = 9 nnd the cone rty



Answers

Use spherical coordinates.

(a) Find the volume of the solid that lies above the cone $ \phi = \frac{\pi}{3} $ and below the sphere $ \rho = 4 \cos \phi $
(b) Find the centroid of the solid in part (a).

Okay, so I want to find this volume between guys Ah pi over three and rose, he could afford co signed by. If we do a little bit of Mathis, we could find the equation of this fear and Cartesian coordinates to give us an image of what it is. And we can, um, find that it's just ah, sadly, the sphere a question of a sphere with radius too, and centred out 002 So we know our images looking like something like this. So we have the beauty space. We have this fear sitting atop text by plane member. It is, too. And we want to cut it with this cone, uh, five pi over three and find this inter volume. So to do that, we know where bounds are. We know what our balance are because this is has circular symmetry is we have Syria to pie and data. We go from zero to pi over three by and 0 to 4 co sign Phi and they in a row and witnesses for just going to volume. We know this is gross squared sine phi the road. If I d data since we have, um, coastline find our bounds. We can just separate the's just yet. So we have to bring, uh, by with the inner integral first, um, giving us four co sign if I cubed over make yeah for co sign Phi Cubed over three. Um defied data and now we can start separating, so we can separate. Um, first can't forget. Um, the sci fi we have They're orphans here to two pi D data and integral from 0 to 2 pi over three. Um, type of 64/3. Time its coastline cube fi over. Cause thank you, fi. Not over. And things that I already have that included in front sci fi. If I this first integral is just two pi and the second integral is 64 or three. We can do a simple use sub you as you go to co signed by and the U is negative sign fighting pie. To find this integral is just none of co signed by Over to the fourth power over four from pi over 30 are zero to pi over three. From this we get the following valuation negative 1/2 to the 4th 0 R. The negative is in front. So we have negative 1/2 to the fourth over four plus 1/4. This is equal to 1/4 minus 1/64 or 15/64. So we have this whole integral values to 15/64 and, uh, with a bunch of cancellation, we see that this is was away. This gives you five. And so our volume is simply 10 pie. To find the central aid, we want to think about symmetry first. So we know that since its circular symmetry that the central it must have, um central. It must have X equals zero Weichel zero. And, um, we have the following, uh, interval itself for easy since ah, it's only the top half of the cornet space. We can't just say cemetery gives it zero. So then we need to find Z. So this service coordinate the average value. Z is just one of her volume. The integral from over this region are of zee TV. So this is the same thing as 1/10 pie integral with the same bounds of ZDTV, which is for ah ro co signed by Times TV or Row Cube signed five co sign Phi dear. Oh, defied data. Now we can go ahead and sell this, um, like before We can't separate the intervals just yet. Yes, we have. It's to outer into girls and center printed role, which evaluates who wrote the fourth over four. So we have for a co sign by the fourth over four sine phi sine phi 85 you data and, um, bringing this thes constants out, we get 64/10 pie. Um, now we can start separating. Integral is to give us here to two pi data and zero to pi over three of who assigned to the fifth power. Fine sci fi defy. Now, um, we get a two pi here, which cancels a two pipe from the bottom, giving us 64/5 times. Um, from, like, before, this is just equal to a negative Kasai into a six. If I over six from pirate 3 to 0 or, um, 16 minus 1/2 of the sixth over six, which is 16 times one minus one over, um, 64 or 63 over 64 times six, which is, um, convenient. Since we have a 64 on the top of the bottom. So we have this cancel and then we're left with 63/30 as rz cornet. So then now we can further cancer this divide by three to get 21/10. So we have the following coordinates. Centrally coordinate 00 21 of 10 and the volume of 10 pie.

For the following volume. Um, he's, uh, or is a bounded between the corn. Um, by this commonly described by he eagles to my hurts between its very goldkorn needs. Very cool. It is. Above all, the next life plane explains this quite lately, saying that Zizek all did you on your side of the on this fear of really usedto So you would have something like that. Sort of like these region. No. So that is this fear of release to So what you say? I love that and also roll small them too. So you like to book you? What is the volume of these? These volume. So, uh, for feed you know that he's gonna meeting by thirds on the single. But I'm going there. Something equals to my house. So, uh, feel beating by thirds. Um, by hops on roll go from Syria to on, uh well, fair goes totally around. Were they not so the whole turn from zero off by two to play. So the volume is gonna be able to, um so fair goes from Jodi by buddies. The bombs 40. She goes from 5/3. I put my house. There is defeat on row goes from 0 to 2. You know, the Devil UMA Women Peace role Sign of feet. This is the volume element. So Marie melamine is very so you need a great, uh, it offers because nothing here depends on finish. He said, uh, from zero. Dubai is just gonna be equal to theta when we did to buy on a zero to would be able to just to buy So all of the central thing that movie go to to buy times drill from by thirds off by house. Oh, literal from 0 to 2 off road squared, sighing the row, the feet No. So, no, the reading throw room. You have no control. A rose clarity grow that would be able to roll cube thirds. So we are doing that. Taking too, and zero you don't pain to with their power hurts minus hero. So we'll be able to that. So we'll have that these people to to buy times Lewis there blower be very by three and then we help the last into Roland Girl from my third something by house Books Idol. He few room for the drill. A file sign draw sign where hes minus co sign Mine is close cause you view the friendship because with respectable See the thing you have fresh with minus call, Sign the pain sign. So, uh, you would be is regal toe these internally will do to buy them stew cube over three times, minus so close off by house, Myers cause by thirds I thirds. Uh, So who signed off my house? About zero. Dana. They have minus minus. So well, we just less well, that comes. We don't know what you're saying. Co signed off by thirds the sequel to together you circle your This is one, Then we'll by thirds. He's like these angle because, uh, he's, like, dividing how I turned out the spy into free free slices so that we're gonna have, like, one toe. So these families, 5/3 I'm doing well, it's cool signs. You will do 1/2. So this is one, uh, nice about, Of course. Why third's So these were the 1/2. So miners might as 1/2 of these 1/2 councils want to there. So you obtain that we should be gold miners. Why? No, that's positive. So good. You have ah positive number which we sure is a volume. So it's gonna be do. Their power comes by thirds. But did your thirties going toe eight eight by third's mom? So that should be It's about me life paid by thirds.

It's fear over you. So a some solids fear X x one What? This scene Thanks. The Z axis The Y It was very nice is fear you say she been We could through the cold described by room free physical toe This Kony's Yeah, I know Be physical too by over four So half of don't up the various by house This is gonna be, uh, you my hopes, my faults How these cone this shape, it's called illusion. You We went to find it centrally. So in Troy, Um so, uh, by the symmetries, this is not a good, respectful fair in we're working this very cool. According Israel. No Farah working so symmetric with respect affair The central day average X was 20 There was wires with your purpose. Oh, it doesn't mind European, the figures things the same. Trying to find what is there that point that is gonna be the average sea farmers. See, that's gonna be see not the sea component for the century which can be computed us one over the value of the well times the average See cavers values e inside, degrading over. Okay, so well, this volume for that volume you can do. Um, it's very good co ordinates to do volume Element is rolled square sign. So your feet now the role the So we'll roll. Well, this is a this goes up to radius a that he's grow within a of being RC It's gonna be it means you're on by were for so for road have, huh? There it goes from zero up to a eagles from Cyril off by over four. I'm theta goes all the way around this year The to buy Very nice. So if you do first need to deal with this big profit, we have this integral. You said, uh, from zero to buy because that seems nothing here depends Olvera variant. Really Just do buddy to back so you can obtain a factor to buy because they dropped from 0 to 4. When we get rid of that internal. This federal income from syrup off road square sign on me. The grow the fee. So they drove over Rose Choir duo these vehicle tow rope cube. Third's. That's a lot of ah located within zero on a you obtain, um to buy them say cuba third's times internal from zero to buy over four most style fee. The he Well, I'm a drill off Sino feet you saw minus goes remember, Uh, so that would be not between other four 01 of equal to minus course of buying before minus cause of zero. Wow. So these because of blubber for he's, uh, squared off to to I was so serious. One one which can be read in us tour to. So by doing these so, structure, we're gonna be leading us that these, uh mine Is that so? That minus two to minus great. The thriller, too, Which we have that minus can be reading us to minus squared off tool all of that water too. So, um, you know that that but volume is he able to do by third's times a cube? I'm still minus X squared off to Albert too. We can cancel 12 there. Um, so these But is it will you? So the volume is people too. These while Younis, if you I'm still minus the square root of two over three on the in them for these, be central. Well, we can dio see. Is he going to roll? Course she would have robots beyond the William Element. Israel Square sign. Well, she So you have that then hero you do There. I'm row goes from syrup A no vehicles from Syria to by over four. What? They're from zero up to buy. So here what we have things will roll Cube the row from zero to a These were people too. Road to the fourth, divided by four So that evaluated between you and zero is gonna be April 4th power. You know, like four friends You something that I'm factor 1/4 you voted for. And you can do natural respect. The federal, because there's no function depending on whether they're so, uh, herbs irritated by this is just people Sarah, the weaker to buy miniseries. Oh, to buy. Yeah, that people to that times to play. That was the central. The drill left is in trouble with respect feet. These in turn from zero off by over four off course we sign. Feel the thing I'm for these You can do. You can do one new substitution. You get these being you. So you musical toe course home. We say that this is aren't you the u legal toe minus sign. We fear the central, uh, would be minus things fell off you. Do you? Because you have There you have been, do you? It's minus. And then we have here for the bonds. For what is your by or four. For what? It's you at zero. So you have by over four is gonna be cause of filer for because of by over four, which she will do, score it off to Over to that's here is gonna be goes off zero cheese one so that this interview is gonna be minus. You squared, you squared hubs on God I've always beating Scurried off to or two on one. Um, sort of these becomes minus scourge of to over to, ah, squared on being a 1/2 of that. My, uh, one square, which is one divided by two. So these number here is gonna be, uh, these are dear is going to, uh, divided by two squared. So these two will cancel under there. These will become 1/4, but it's a square minus one on these. Uh, what is it with these? This 1/2 is able to do. We're the square, which is for so these will be able to minors. So minus one witness through his mind is one over four of his minus minus 14 That is 1/4. So these, um these are me 14 So that this whole thing will is well for that. A leasehold internal or the paid to the fourth power them strew pile. Very but four. That was 1/4 these waas that interval Mrs. Ah deriding. See So that for CEB are you need to if I that internal by the volume So that Steve our vehicle toe never see which is the sea component for the century Javy that number 1/4 them still by right Abi for them. Spore divided by the volume that the volume musical dough that divided by Okay, que we have a three on the bottom on two squares off to that is a volume and then all these can be regionals. Ah, I want to play bad. My dad You want to play done by that? Uh oh. Sorry. It was a fact off by So yeah, that she was people toe paid to the fourth. That's too fine. Times three divided by four times four times by them's The Cube. That was still, man, is this quarter so that these becomes, um, these councils three boys, there's we get up or a one Partovi. Then I'm being by councils on the other three of these councils to there's a would be equal to anytime Spy times three over to them's for Who knows? For I'm still minus X squared off to businesspeople toe three a by over it tu minus is great. Um, so that is gonna be well, that's gonna be, uh, the value over the central, the secret in for the century, that's, you know, until that

In part a where asked you spherical coordinates to find the volume of this solid that lies above become Phi equals pi over three and below the sphere grow equals four co sign five First of all, notice that really grows for co sign five This implies that we're squared is equal to four road The sign five This means that this is the equation of a spear of radius to with the center you're looking at our second equation The centers Cuando lie at zero zero to and therefore the volume of the sphere is the integral from 02 two pi integral and we're actually integrating below This fear really equals for Kassian five above the coun bye equals higher than three and we're integrating from zero to pi over three integral from zero to grow equals or Kasan five of one times the differential which is gross squared signed by zero. Define fatum and taking into derivative with respect to grow just in the growth of 0 to 2 pi integral from zero to pi over three uh, one third world cute from row equals zero to really close. Four co sign five are and times signed data decide you say that evaluated you get integral from 0 to 2 pi integral from zero empire of three of 64 3rd co sign cute fi times Sign five. The five year data can use the Venus serum or simply take anti derivatives with respect to fine good girl from 0 to 2 pi times I'm doing you Substitution in the head becomes negative 14 times 64 3rd. So negative 16 3rd co signed to the fourth bye from five equals zero five Close pi over three Did data evaluating We get the into role from 0 to 2 pi of negative 16 3rd times 1 16 minus one Data simplifying and taking the impact riveted. This is five times data from 0 to 2 pi, which is 10 pot. This is our volume in part B were asked you spherical coordinates to find the Centro oId of the solid from party. So for the central oId, we're assuming that the density is constant and is one now Notice also if you envision our surface Sorry are solid. You have that by symmetry. Well, it's symmetric about the y Z plane, and it's also symmetric about the X Z plane. Both of these moments about the Y Z and X planes of both zero. And we have at the moment about the X Y plane. This is the triple integral, which is integral from 0 to 2 pi integral from zero to pi over three. Integral from 0 to 4. Co sign five. The party scene is about of them. Um Z, which is road times cosign by times roast bread. Sci fi. This is real cute. Cosign five signed by zero Fine. The data The Z come to differential in spherical coordinates taking into derivative with respect to row. In evaluating, he gets his role from 0 to 2 pi each broken zero pi over three. Uh, maybe it's co signed by signed by times. See 14 funds order. The fourth is four cube of 64. Co sign the force. Fine. Five. This data and taking a derivative with respect. Fine. You get into growth from 0 to 2 pi uh, during a U substitution in your head. This is It was negative 1 50 times 64. So I'm sorry. 16 10, 64. Maybe it's 64/6 times Co signed on sixth Fine from five equals zero to fight equals Hi. Of the three data evaluating, we get integral from zero two pi. Uh, higher the three. Right? One harm six. Uh huh. Of course. 4/6. This becomes 21 halves. The data taking the anti derivative and evaluating this is simply 21. Hi. Right. This is our moment about the X Y plane and therefore the centrally export. Why? Barzee Bar coordinates zero zero and 21 pie over 10 pies. 21 10th or 2.1.


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