5

16. (4 points) The volume of the region in 3-space given by the condition 22 +y? < 22 < 12T (A) 3(B) 3 (C) 6, (D) %, (E)...

Question

16. (4 points) The volume of the region in 3-space given by the condition 22 +y? < 22 < 12T (A) 3(B) 3 (C) 6, (D) %, (E)

16. (4 points) The volume of the region in 3-space given by the condition 22 +y? < 22 < 1 2T (A) 3 (B) 3 (C) 6, (D) %, (E)



Answers

Use the best method available to find each volume. The region bounded by $y=x$ and $y=x^{2}-6$ revolved about (a) $x=3$ (b) $y=3$ (c) $x=-3$ (d) $y=-6$

In this problem, we have finding the volume bounded by the equations y equals X it before and why equals to the coastline of X. First, the region will be rooted about the line. Why equals two? No, this is a vertical line and formal equations. We can do ex integration. So what that tells me is that we can use the show method because we have X integration about a vertical axis. So here with Williams the show Vertical show method No recall. But this would need to the equation representing the radius on the height so we can find a radius by tu minus under height you can represent by cool sign of it our opera equation minus the Louis question. Exit 1/4 to sit up the integral we have two pi the body musical to two pi times the integral ah ex going from negative point 891 to positive point H 9 11 If you look at the graph, you will see that the points of intersection occurs at those X values and we have the radio. It's to minus times the height. We'll sign of back minus next it afford time. So what okay. Using the calculator to evaluate the integral, we have two pi times approximately two points six to which gives us an approximate volume of about five point 3 to 4 times. By part being we were treating about the line y equals two. This is a horizontal line with X integration. So we will be able to use the washing method here since I was started Will have her coveting. So we used the Washington Wash your hair. Okay, so we need to identify the outer radius. I'm squared and we can do that by Tu minus only question exit the fourth under in a radio, we get by attracting coastline from two total volume will be found by pie times the integral from X equals negative 0.891 to positive 0.8 91 times. Oh, alter radius squared. That's too minus extra fourth squared minus the inner radius squared. And this is ex integration. So this gives us approximately 4.0 to to high as the volume for part C. Rotating about the X axis is the horizontal line and we don't x integration. So again, we will use the washer method here. Okay? And first let's identify the outer radius. And that would be call sign of X minus zero, which is just school Sign of X. Nothing makes clear. And oh, it in a radius would simply be the equation extra fourth squared, which is just excellently. So the volume being would be given by hi into girl from negative 0.891 two X equals positive 20.891 about all tour idiots. I was saying squared act minus in a radius squared and that works out to be one point 301 sky one of volume. Now we rotate above the y axis that is a vertical line and we're doing X integration. So we are able to use the show method here. So we need to know identify our radio under height. So really, it's our with the X men and syriza this clear Under height h, we can subtract extra before from Khost sign of eggs. So for volume, we have the question. We is equal to two pi times the integral from negative point 891 to positive Point it 91 Okay, If we multiply the radius with the height we end up with the inclusion X I'm school sign of bags minus exit If it the X now it's important to note. Here we have an odd function. So if we take the integral from negative 0.8912 positive for 91 we will get an answer of zero. Because the volume a book the Y axis is equal or to the right of the Y axis is equal and opposite to the volume below the X axis. So to account for that, we can we write our equation just double, huh of the integral. So this would be to pie times two times the integral from instead of negative 0.891 will go from zero to positive. Point it. 91 Mr Lee, Both sign it minus x it a bit the Okay, So what we end up with is for pie times approximately 0.238 from the integration, which gives us a volume of approximately point nine by two pie

Marie have these reasons that he's being was bounded by Yeah, X see waxes. So it is bounded below by the cone. This colder Sample these five boards. So these defines a cone. She was one of the scone. We're here. It's below that Cone Indian. No, it is also supposed to be inside of this year. The radio's on three. So it is, uh, this vision inside. So, uh, I'm boned, you know, Lady Cone, huh? Equals through my faults. You sigh well, missing. They're off this year. Ovary already. These three. So you have these boys in D? No. Well, I mean, it's very cool. Cornelis circle? No. So that bones for she, uh, he must be between. So, between zero, we're four days for, uh, Rose between you're on three cereal three. And then, um there goes all the way around. So whole turn from zero to Dubai. It's your goodbye level element in Sprinkle coordinates. His row squared sine wolfie to the volume during this because the volume by the So now we wanted to do things boil you in. It's, you know, people, Gordon. Until we have their restrictions that are well, you have here the X, the seaplane. So, uh, maybe he's, uh, over. These is going on inside That, uh, this few of ages three. Um, also, the restriction is nothing c square plus R squared. Well, must be smaller. Done, huh? She Yeah. I'm being, uh, so o r please. Our It's, uh, was vying from here to there. There, there, there, there. So this is their projection. You should be the positional farm. So this is the line. See, people's toe are But is that gives you Yeah. Why? Works. So are her Really use will be between, uh, zero on and, uh, see. But we have this condition that well bound stops being good at this point. That point has something not that is That would be all these long years. Three. So that is seen not wishing the three times sign. Well, you are my fourth. She's, uh, three. Everybody squared off two. It's quite of Dewey's one of Earl's court of Please sign off my fourth fine forth. So you should do, uh, Well, this bond is good for for C. It is. You on, uh, that that number Duffy, Where is good to know, But you should be you are is gonna go from zero up to see So they're gonna be relenting, Cylindrical. These are our undying, uh also the sea. Uh uh. Zeros from zero off. Do. There's no number. You ever squared off too, when they indeed theater, uh, goes old, we're on. So it's like 13 these rooms, the oracle. But then you have to do after plans. Either. Another needle, uh, which you have the bone for. See you eating three squares of film from that number. So there the bond. Versi. He's even by that. So that you simplify that together. C o the bone for our do you have when she's three minus c squared fleeing When is he squared? He's goingto are square e r. We'll go between zero and squaring the three minus c squared. So the bone for our should be It is zero on the ground. No, she squared Radios think so. Three square minus. Is he squared on dinner? Ovaries? This is the bone. For our there is a lot of mercy on being thorough has can go hold the way around so is easier to. So the volume should be able to those two intervals I did in Cylindrical. So Ivy's easier to complete it in. These gets one. So wherefore dog off things go from zero two guy four squared sidle he the p and then rose from zero off the three free on there. Hey, I goes all the way around. Yeah. So do it, huh? Do a thief eat first. So they drove. Sign what will be minor school sign. So you want it bad in 54 on zero or this is gonna be miners like Ford's minus close. I know. Do you mean? Well, was in my forties. Whatever score of two is one. So mine is That would be hotel. Well, these, uh, these one can hear it in us. Uh, do ours too. So you have minus one minus one minus one one minus squared off. Or is going to do so. These numbers, it's quarter to ah, minus one over. Squeals off. Do No, That is this First for the people, too, too Screwed up to my as one. We were escorted through, huh? Things interval with Inspector Federer into old respect too. Row from 03 Scored the road there. So will they go? Angel bro squared. You don't kill your old Q. You're like three. So that the awful waiting for people to two to his one device placed too times. Uh, three Q If I would like three. My nose. Uh, well, your cube got away free. It is you, Rhonda, All times these interval. Bye. Oh, the fear. Uh, So before moving on, let's make some simplification. See these as those Weren't power so useable conserving that would be the square. Right? So we have that volume's you see, Phil two for the two might is one over square with two. There's three squared. That's three square. Sometimes Inderal from zero to buying. Ferre is central visible to do. Bye. So that the vehicle to hole seeing Yeah, that would be excludes a tool in his one. Yes, that's nine. Because mind them still, there will be 18 bye over square. So that is so where is avoiding on these of these? Well, that region

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

In part a were asked to find the volume of the region bounded by parable. Lloyd's Z equals X Squared plus y square and Z equals 36 minus three X squared, minus three y squared. Well, you have these two parable. Lloyd's intersect when their values of Z or equal So when X squared plus y squared is equal to 36 minus X squared minus y squared. And so we obtain that X squared plus y squared is equal to sorry. It's three x squared, minus three y squared. So extra plus y squared is equal to nine into the region of integration. D is the set of pairs x y such that X squared plus y squared lies between zero and nine. And in cylindrical coordinates, we have that our region. Mm. Well, this is the set of triples our fate a Z, such that data of lie between zero and two pi, as usual, are is going to lie between Well, we have X squared. Plus y squared is less than or equal to nine. Supplies are squares. Listening to the nine that are is less than to go to three. So our lives between zero and three. And finally Z is going to lie above the upward facing parable Lloyd, which has equations equals R squared and the downward facing probably which has the equation 36 minus three times are squared and therefore the volume of this region. The is the triple integral of one over e, or it's the integral from 0 to 2. Pi integral from r equals 0 to 3 integral from Z equals R squared to Z equals 36 minus three times R Squared of the differential are dizzy DRD data breaking into derivative with respect, dizzy and evaluating, we get integral from 0 to 2 pi integral from 0 to 3. Then we have 36 are minus three are cube minus 36 Sorry minus R a Q introduces to 36 are minus four r cubed, he argues Data music for beans theorem weaken right. This is a product of into rules or integrating directly. We have integral from 0 to 2 pi um 18 R squared minus part of the fourth r equals zero to r equals three and evaluating this gives us integral from 0 to 2 pi um 18, 10 9 minus read to the fourth, which is 81 deep. Data taking anti derivatives and evaluating this is 162. Hi, in part B for has to find a century of the region e So we're assuming the density of this region is constant. We'll call this constant density K and we have that the total mass of the solid am is K density times the volume of the solid which calculated in the previous problem This is 162 pi times some constant k. Now our region is homogeneous and symmetric. Homogeneous just means that the density is constant throughout and it's symmetric with respect to the X Y plane. So we have the moment about the y Z plane is equal to the moment about the X Z plane, which is zero, and we have the moment about the X Y plane. Well, this is, by definition, integral from 0 to 2. Pi integral from r equals 02 r equals three integral from Z equals R squared to Z equals 36 minus three r squared times z times the density K in the differential are Deasy DRD data and taking the anti derivative with respect to see we get que times into growth zero to pi integral from 0 to 3. Memphis is rz from Z equals r squared to Z equals 36 minus three our spirit DRG theater. I'm sorry. It's wrong. Should be our times one half z square because we already have power to see that. So evaluating we get okay and then pulling out the one half que over to he was integral from 0 to 2 pi integral from 0 to 3. Um, our times 36 minus three r squared, squared, minus r squared, squared were art of the fourth DRB data and using food Beanies the're, um, and multiplying things out you can write. This is a product of in Jewell's this is K or two times integral from 0 to 2 pi d theta times integral from zero three of And here we have that the exponents for art of the fifth is a There is no exponents for art Of the fourth, the explosion for art of the third is negative. 216 r cubed and the exponents for r squared is zero and the exponents for our is 12 96 we are taking anti derivatives. We get cave or two times two pi times four thirds art of the sixth, minus 116 over four. Part of the fourth plus 12 96 over to R squared from 0 to 3. Evaluating we get I times K. Then this whole expression here reduces to 24 30 and rearranging This is 2430 pie. Okay? And therefore it follows that the center of mass X bar y bar. See bar. This is the point with coordinates the moment about the Y Z plane over the total mass moment about the X Z plane over the total mass and moment about the X Y plane over the total mass. Our previous calculations the first two components are zero. Third component is 2430 pai que over our total mass, which was 162 pi. Okay, in simplifying, we get zero zero 15 as a center of mass


Similar Solved Questions

5 answers
(6 pts) Graph y = In (x+ using transformation Please label at least TWO points on each graph including the x-intercept and any asymptotes(s) If you use point-plotting or fail to use transformations in the correct order; you will earn ZERO points on this problem_ Basic function: Describe first transformation: Equation after first transformation:Describe second transformation: Equation after second transformation: Y = InG*+1)(4 pts) Given 2x - 1andg(x)Find
(6 pts) Graph y = In (x+ using transformation Please label at least TWO points on each graph including the x-intercept and any asymptotes(s) If you use point-plotting or fail to use transformations in the correct order; you will earn ZERO points on this problem_ Basic function: Describe first transf...
5 answers
Drartink the then Newman projections continuing H,c V 2 H carbon-carhon puoqfollowing molecules
Drartink the then Newman projections continuing H,c V 2 H carbon-carhon puoq following molecules...
5 answers
1a. Let Find AnB 1 and B = {r € Nl1 A <10}.b_Find B - A1 4 three of these properties are
1 a. Let Find AnB 1 and B = {r € Nl1 A <10}. b_ Find B - A 1 4 three of these properties are...
5 answers
Use double-angle formula find the exact value of the given expression:1-2sin 267.5?1-2sin 267.58 (Simplify your answer, including any radicals . Use integers fractions for any numbers in the expression:)
Use double-angle formula find the exact value of the given expression: 1-2sin 267.5? 1-2sin 267.58 (Simplify your answer, including any radicals . Use integers fractions for any numbers in the expression:)...
3 answers
End ononeceoler they eavt Hlice < 6Smilest 1 Fino -cetaycrole speen: (Roundyou 1 niorth 1 Jnswch eaneselemia pajds Jiljjal $ mLpI Sloweritnan Ancz
end ononeceoler they eavt Hlice < 6Smilest 1 Fino -cetaycrole speen: (Roundyou 1 niorth 1 Jnswch eaneselemia pajds Jiljjal $ mLpI Sloweritnan Ancz...
5 answers
Question 1ptsGiven the basis 8 = ((1,2) , (-1,2)) what is the coordinate vector of (2,2)?(2,2)(-1,2) (3/2,-1/2)(1/2,-3/2)Question 21 ptsIf 8 = (e3, = e2, e1_ and [v]8 (1,2,3) what is U(3,2,1)(0,0,1)(0,0,0)(1,2,3)
Question 1 pts Given the basis 8 = ((1,2) , (-1,2)) what is the coordinate vector of (2,2)? (2,2) (-1,2) (3/2,-1/2) (1/2,-3/2) Question 2 1 pts If 8 = (e3, = e2, e1_ and [v]8 (1,2,3) what is U (3,2,1) (0,0,1) (0,0,0) (1,2,3)...
5 answers
"9560 with Exactly standard deviation of 0.0020, normal wlth 4 = 9560 = wnal A0 and: samping Approximately normal 0,0020 dlsuibur Exactly normal wChe = 9560 and & = 00020 wlth / 9560 and Approximately normal with 0.8k920 /v33 . p=,9560 and 0z 0.0020 /33(c) What theorem did you use to answer part (bJ?Central Limit Theorom Chebyshevs Theorcm Pythagorean Theorem Law of Large NumbersMc Graw HillPrey2 0f 24
"9560 with Exactly standard deviation of 0.0020, normal wlth 4 = 9560 = wnal A0 and: samping Approximately normal 0,0020 dlsuibur Exactly normal wChe = 9560 and & = 00020 wlth / 9560 and Approximately normal with 0.8k920 /v33 . p=,9560 and 0z 0.0020 /33 (c) What theorem did you use to answe...
5 answers
Perform the indicated operations.$$9 rac{3}{7}-2 rac{13}{21}$$
Perform the indicated operations. $$9 \frac{3}{7}-2 \frac{13}{21}$$...
5 answers
Find the power series expansion through the uniqueness theorem(x = 0) for:(a) cos(x)(b) tan(x)(c) sec(x)(d) exDo not use taylor series approach
Find the power series expansion through the uniqueness theorem (x = 0) for: (a) cos(x) (b) tan(x) (c) sec(x) (d) ex Do not use taylor series approach...
5 answers
Please submit one page ONLY paper on the ketogenic diet (what is it? would you recommend it or not and why)
Please submit one page ONLY paper on the ketogenic diet (what is it? would you recommend it or not and why)...
5 answers
Draw the structures of the Azo dyes you would use to make a blueand gold t-shirt and include the necessary pH conditions.
draw the structures of the Azo dyes you would use to make a blue and gold t-shirt and include the necessary pH conditions....
5 answers
Use 4-1 to decode the cryptogram Enter the decoded message below:)A =368100151195346
Use 4-1 to decode the cryptogram Enter the decoded message below:) A = 368 100 151 195 346...
5 answers
5 cot 6cott + Pcove tt 2) Lot 4+5 Gos?x - Sin?x
5 cot 6cott + Pcove tt 2) Lot 4+ 5 Gos?x - Sin?x...

-- 0.059225--