5

5 points) A solid is bounded from below by the cone 2 Vz? +y2 and from above by the plane 2 = 1. The density of the solid is given by fr, 0,2) = 22. Find the mass o...

Question

5 points) A solid is bounded from below by the cone 2 Vz? +y2 and from above by the plane 2 = 1. The density of the solid is given by fr, 0,2) = 22. Find the mass of and the average density of the solid.

5 points) A solid is bounded from below by the cone 2 Vz? +y2 and from above by the plane 2 = 1. The density of the solid is given by fr, 0,2) = 22. Find the mass of and the average density of the solid.



Answers

Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. The solid bounded by the cone $z=16-r$ and the plane $z=0$

So for this problem that we have here, we want to find the center of mass eso to find the center of mass. What we have is e equals X squared plus y squared NZ equals 25. So we know that the region is symmetric about the Z axis and the density is constant. So based on that, we see that X zero and Y zero and that's the location of the center of mass. So we know that the location of the center of mass is X y z Andi, that's equal to M y Z over em m x z over em m x y over em. So to find mm y z, this is going to be equal to the triple integral of X d V. And then M X Z is going to be the triple in a role of why Devi and then similarly, M X Y is going to be equal to the triple integral of D of Z TV. And then lastly, what we have is that the MASS M is equal to this right here. So with all these things in mind, we want to use cylindrical coordinates and what will get as a result is since X squared plus y squared equals 25. Since this could just get substituted, we see that are squared, equals 25 artist equals five. So that's how we can change the limits of integration. Eso we're gonna use our instead of Z. So with this logic, we see their bounds of integration for here 0 to 2 pi 0 to 5 r radius and then r squared to 25 on what we end up getting as a result of that is going to be 6 25 pi over to and then using similar bounds of integration will use the same bounds of integration on DFO for MX. Why we're gonna end up getting 15,625 pi over three again using the same bounds of integration, just changing the values from Z. Based on that, what we end up getting is we can take this value right here and divide it by this value, so that's m X y over em. And we know that's going to give us 50/3. So based on that, since we already knew X and y, we know that the center of Mass, which is X Y Z is going to equal 00 50/3, and that will be our center of mess.

Very nice. Solely sorry. Off, uh, constant density. Constant Lisicky on. Uh, it is bounded by the plane c equals zero the year x y z The plane Sequels Here is the X Y plane. He responded below, Like that plane on the top by the cone Our physical to see So your piece round comb Are these people to see, um on duh on the sites by the ceiling There, over his one about the C axis. So you have these, uh, very nice volume it is then interior off these so something that the I do want to find center of Must Santer off must so well on center of Masisa the average well, uh over each hornet is gonna be They are very self eggs. We are so the original Mix the aberration. Why garish z? So that's gonna be the center of loss. Our or this is symmetric with respect. The rotations. So, Fada So that these two guys, they I'm Irish Exxon over his average y will be zero to the center of mosque. Most light in the along. The secret is gonna half court in it. 00 some number. Do you know I'm gonna be the center of mosque you need to find what is. That's not That's not is a average body. You'll see. Would she have been computed of some the general over? Deal off the value off. See X Y Z over from the volume off the also first books compute. Why is that what he's got? So I think this region is, uh, in cylindrical coordinates. See you, Rico. The volume is gonna be well. We do a par, is he our? So, as you can see him, destroying on a boat from from zero up to are is the line. See, People are so ce goes from zero up to are but you can not this year are fire goes up to one the bodies one So are from 0 to 1. I'm Zoraida. He said that I would go all the way around so 0 to 2 pi So this would be the volume and then a Or be doing that your first internal you see it is your honor are just gonna you'll see polluted our so it's gonna be are zero, you know factor Ardor? No, not that. Into lives in the vehicle to our them sarsalari square er from 0 to 1. De from 0 to 2 pi. Um, so the drill a bar square? Yeah, bars Courtyard does it B r q 2/3. So that of all of it is eating 10 he's gonna be 1/3 zero. So you have that is gonna people to want third over, control from zero get to buy better. So, uh, or here that would just be Ferre Bolliger to buy my zero. So I would be I used to buy So these Morley was gonna be able to to buy third's. What is the volume? He's one. So, um, over the central enough, everyone is the same with you. Have toe lugging their a C She would have been in trouble. See, that's our You see we are Do you think on the same bones Segal's from job to our higher goes from 0 to 1. Ferre from zero to buy. It was a tradition to go. Now we're into reading, seeing this year she's gonna have won t Here are C square hubs so that is severely did that begins here on Are things gonna be our square house in my cereal So we're gonna up Dana on our square house. So we have all right square hops. Times are we are not there until about your Argos from job to one, from zero off to buy, and so over certain doing that would be integrating well, so are cute halves he are. So the intro of these have the you are to the fourth later. But for now we have ah to there. So it would be that what I didn't serum on one. So you nothing. One we're through and score my cereals of this would be one age. So I think that that was one a all metro from zero to buy this intro just there to buy. And zero said you got to play sort of. These is equal to how that too by your body by eight. So you have that This is 25 I ever ate Sudeep average. Look. See? You know, equal to Dubai divided my age and that divided by they tell volume that is equal tools Dubai thirds to by third's We give these feasible to buy. I have three delighted by a tends to buy, so to buy names. Three like a past two, but she and very nice. He's a people to, uh, three eights. So that's gonna be the center of mass with these this, uh, figure out like that she aids. So? So for that, I want to be in. They see gold in it off the center of months. Javi creates set up center of muss. Smooth blast is gonna be zero zero. You're for extra over white and then three seats for Z is in partition. Cartesian coordinates 003 It's

That's solid for cause Santa the city on, uh, so build. We would like to find that we can just make it one. Uh, the diesel. It is, uh, bounded by the plane is of all of the plane c equals zero. So that plane is this playing here? X Y plane is bounded on the top by the cone. Are you close to see? So this cold in cylindrical corners coordinates describe why are equals to see. Are you call strong? I see on DA by the Syrian there are equals one on the sites. So inside of the seating there, over these one. Ah, so the ceiling there gonna be something like that? These? Yeah. So this is the ceiling there. Ceiling there. Well, are getting by r equals one. So it's gonna be he's inside region, the inside of these region. Do you want to find a center of must center? What? Uh, so these, uh, for these would be sorry? It's called, uh, news. The Saudi border. Well, that no more in cylindrical coordinates is gonna be well, see goes between. So yeah, here, C r you notice era. Oh, well, it our go from 0 to 1 c is gonna be ranging between zero are she's gonna go between Syria are brown. Are those from Syria off one on the all the angles. Vera is free to being away around. So fair I would from Syria to buy. This is in cylindrical coordinates. Um, so this center, by the symmetry, with respect to the sample Clara, the center most line in the origin on the X Y plane. So the center of mass will have x coordinate. So these should be equal toe. The average six should be going to the average wife. Why, we should be zero by this unitary. So we need to find, uh, What is the average sea on these can be computed us one over the volume. Well, for this and then that times the interval. Hold her. Yes. Off you are. You will see holding. This is a differential volume. This is in the little girl. Coordinates are see the are the fear his differential over volume. So, uh, and they really more face he's gonna vehicle to just do it the whole So they should be going to being the role. So are you see are the theater where CEOs from syrup to our are those from Syria warm Something loose from Syria Could buy un believe it Here are Well, they should be the same. Um, so let's go. First on compute these volume. Well, you did great. See, Jill, there is this constantly respects. Usually PC is asking to see so data below uniting our zero is gonna be, um, additional front through a bar. So you would have, um Oh, we left the danger. Mark's square you are. Here are from 0 to 1. Get up from zero. Wanted to buy. So this would be the jobs are square physical to our Q tits. So do the equal to arm one so that what we did at 10 would be 1/3 of the times enter from zero to finding. Well, give to buy for the volume. Should be 2.6 is the volume. How s no, uh, lease is gonna be well enough. Seem limits from a zero. You are Soon. He are. Are those from Syria off? One there goes away by being a, um we have another from the volume element. And see, So here we have to integrate. Well, see, you're all seemed deceit. People to see squared halves and so bad I've always did became cereal on her. It would be, uh, squared house to get r squared. Alice are are better. So who would be doing so we would be doing angel bomb. Mark Cuban. So in trouble. Bar Cube, G R. He's part of the fourth power by four. That's about it. It is. You won, huh? Is that he was just 1/4 serving a pain. 1/4 from there, ending with this half to top. These should be equal to Yeah, I heard. And then from these you thing 1/4 full, and then you have federal, you know, offs from zero to buy. And there on these interesting is equal to to play. So this should be able to play two times four. And these two cancels. Ah, So these parties and I just buy over four, so the others see he's gonna be by four. Do you know you buy them or you buddies? All right. Is thes body to fine sex So that the one we divide these sequel to multi plane these them. So these things that over the sins that. So that is the three fine over who comes. Four or eight. That's foreplay. So the average seasonal people go through is to the center of Mars, Senator, unless he's gonna be located. Ah, Cyril. Cyril Landing three. Yeah.

In this problem. It's given a solid region in the 1st 10 is founded by the coordinate plans and the Plan X Plus y plus that equal to and the dynasty of the solid is to X and has two requirements. 1st 1 To complete the mess off, the solid and the second requirement toe complete the center of mess. So get e the mass of the soldier. MM will be equal to integration from boundaries off X, which is 0 to 2, and the boundaries will fall. I will be bones of why the zeal to two minus X and the bondage old said, is to minus X minus y four. The dynasty, which is two x. I want a bloody D's. Add deRoy the X. It will be equal to with the integration from 0 to 2 and 2nd 1 from 0 to 2 minus x four eggs minus two X squared miners two x y Do you wind? The X equals to integration from zero tell to four excess Cuba. You're right again. Execute minus four excess square bus for X, the ex equal for over three. So now mass equal for over three and we need to get played. The second requirement, which is the center of mass. So you really start. Boy mm X Roy should be equal integration from 0 to 2 on second dictation. Same as before, from 0 to 2 minus X and the last one from zero to toe minus X minus y for the dynasty to X. Is that not the boy? He's ent Dear Boy D X, which is equal to of integration from Seattle to toe from legal to two minus X or ex out by two minus X minus Y old boards, too. U Y E X, which is equal to integration from 0 to 2. Thanks multiply to Linus X over three. Although three D eggs, which will be equal, tow it over 15. Now we can get mm X Is that from cemetery? It should be the same as M X Y 8/15 and for M. Why said should be equal toe integration from same as before. From zero to toe. Can the integration from zero 22 minus X and the last one from zero to minus X minus y for on Bondi to X is that these are D. Y? D x Mmm. And, uh, it will be equal toe double integration from 0 to 2 and from 0 to 2 minus x forced to x square. Multiply two minus X minus Y do you worry? D x equal to integration from zero toto for two x minus X squared. Oh, or two the ex. And this should be equal to six team over 15 from the Given that we can calculate, export should be equal for over five. And why Bush will be equal to that bar. You two cemetery and it will be equal to over five. Thank you.


Similar Solved Questions

5 answers
[0/2 Points]DETAILSPREVIOUS ANSWERSLARSONETS 3.7.002_Assume that x and y are both differentiable functions of t and find the required values of dyldt and dxldt: 4 = 2 (1? 3r ) (a) Find dyldt when X 2, given that dxldt dyldt (b) Find dxldt when x = 8, given that dyldt = dxldt Enter a fra ction, integer, or exact decimal: Do not approximate Need Help? Read It Talk tto [utorSubmit Answer
[0/2 Points] DETAILS PREVIOUS ANSWERS LARSONETS 3.7.002_ Assume that x and y are both differentiable functions of t and find the required values of dyldt and dxldt: 4 = 2 (1? 3r ) (a) Find dyldt when X 2, given that dxldt dyldt (b) Find dxldt when x = 8, given that dyldt = dxldt Enter a fra ction, i...
5 answers
Evaluate using integration by parts or substitution. Check by differentiating:9x 4x e dx
Evaluate using integration by parts or substitution. Check by differentiating: 9x 4x e dx...
5 answers
The solution of $x d / d x=2 x+y$ is
The solution of $x d / d x=2 x+y$ is...
1 answers
Calculate the molar solubility of $\mathrm{Cr}(\mathrm{OH})_{3}$ in $0.50 \mathrm{M} \mathrm{NaOH} ;$ $K_{\mathrm{f}}$ for $\mathrm{Cr}(\mathrm{OH})_{4}^{-}$ is $8 \times 10^{29}$
Calculate the molar solubility of $\mathrm{Cr}(\mathrm{OH})_{3}$ in $0.50 \mathrm{M} \mathrm{NaOH} ;$ $K_{\mathrm{f}}$ for $\mathrm{Cr}(\mathrm{OH})_{4}^{-}$ is $8 \times 10^{29}$...
5 answers
Question 2Time left 0.59.19Not yet answeredMarked out of 1.00Flag questionWhich of the following is the main kinetic product of the Diels- Alder reactionOPhco;"eOPhOpn NCOMcco_MeIIb_ IIICI"co "
Question 2 Time left 0.59.19 Not yet answered Marked out of 1.00 Flag question Which of the following is the main kinetic product of the Diels- Alder reaction OPh co;"e OPh Opn NCOMc co_Me II b_ III C I "co "...
1 answers
Write down expressions for $K_{\mathrm{sp}}$ for the following ionic salts: (a) AgCl; (b) $\mathrm{CaCO}_{3} ;$ (c) $\mathrm{CaF}_{2}$.
Write down expressions for $K_{\mathrm{sp}}$ for the following ionic salts: (a) AgCl; (b) $\mathrm{CaCO}_{3} ;$ (c) $\mathrm{CaF}_{2}$....
5 answers
Make the given changes in the indicated examples of this section and then find $d y / d x$In Example $3,$ change $x y^{2}$ to $x^{2} y$
Make the given changes in the indicated examples of this section and then find $d y / d x$ In Example $3,$ change $x y^{2}$ to $x^{2} y$...
5 answers
Find the first and second derivatives of the functions.$$p= rac{q^{2}+3}{(q-1)^{3}+(q+1)^{3}}$$
Find the first and second derivatives of the functions. $$p=\frac{q^{2}+3}{(q-1)^{3}+(q+1)^{3}}$$...
5 answers
How many mL ofa concentrated 0.02 M ibuprofen (aq) solution should be added to a 100 mL volumetric flask (subsequently topped up with water to the 100 mL mark) to create a 0.003 ibuprofen solution?
How many mL ofa concentrated 0.02 M ibuprofen (aq) solution should be added to a 100 mL volumetric flask (subsequently topped up with water to the 100 mL mark) to create a 0.003 ibuprofen solution?...
5 answers
Consider Evita and Tariq's Re-cycle business in Problem Set D. Make appropriate changes in their assumptions so that the business has a negative net profit over the 5 -year period. Defend each of your changes by explaining why it is plausible.
Consider Evita and Tariq's Re-cycle business in Problem Set D. Make appropriate changes in their assumptions so that the business has a negative net profit over the 5 -year period. Defend each of your changes by explaining why it is plausible....
5 answers
Explain how macroevolution is predictive. Give two examples ofmorphological species that support this fact.
Explain how macroevolution is predictive. Give two examples of morphological species that support this fact....
5 answers
Exercises 7 - 14 Find $a_{5}$ and $a_{n}$ for the following geometric sequences.$$a_{1}=9, r=-3$$
Exercises 7 - 14 Find $a_{5}$ and $a_{n}$ for the following geometric sequences. $$a_{1}=9, r=-3$$...
5 answers
What are the molecularity and rate law of the elementaryreaction below?NO3 (g) + CO (g) → NO2 (g) + CO2 (g)Select one:Unimolecular, Rate = k[CO]Bimolecular, k[NO3]2Unimolecular, Rate = k[NO3]Bimolecular, Rate = k[NO3][CO]Tetramolecular, Rate = k[NO3][CO][NO2][CO2]
What are the molecularity and rate law of the elementary reaction below? NO3 (g) + CO (g) → NO2 (g) + CO2 (g) Select one: Unimolecular, Rate = k[CO] Bimolecular, k[NO3]2 Unimolecular, Rate = k[NO3] Bimolecular, Rate = k[NO3][CO] Tetramolecular, Rate = k[NO3][CO][NO2][CO2]...
5 answers
22 Use integration and trigonometric substitution to find the area enclosed by the ellipse: M1. 36 25
22 Use integration and trigonometric substitution to find the area enclosed by the ellipse: M1. 36 25...
5 answers
5) Idcntily thc mst stable carbocationAc_=CHC4: 0CH:CHCA:AC==Ch0ChctChch7) Identily thc correctly drwn JiToW >Ch- CH= Ck_84;Ch=Ch_Ch=0h4CkCh_C4t:Ch"CH_CHSFO=C7CCASCA_CHFOC- CHech 04CH Ch_CH_=O4Ck- CECH_Ch:CHS=CH CKS=O+
5) Idcntily thc mst stable carbocation Ac_=CHC4: 0 CH: CHCA: AC==Ch0 Chct Chch 7) Identily thc correctly drwn JiToW > Ch- CH= Ck_84; Ch=Ch_Ch=0h4 Ck Ch_C4t: Ch"CH_CHSFO= C7 C CASCA_CHFO C- CHech 04 CH Ch_CH_=O4 Ck- CECH_Ch: CHS=CH CKS=O+...
5 answers
9. [~/1 Points]DETAILSLARLINALG8 5.1.021.Find the distance between u and v(1, 2, 0), v = (4,d(u,V)
9. [~/1 Points] DETAILS LARLINALG8 5.1.021. Find the distance between u and v (1, 2, 0), v = (4, d(u,V)...
5 answers
Let X be topological space: Two points x and are in the same path component of X if they are connected by Path in X (or equivalently if there exists Path- connected subspace A of X containing both of them): (a) Find an example of a topological space with two Path components and another with three Path components (you do not need to prove they have the desired property) . [4 marks] What are the Path components of Q € R? Justify your answer [8 marks]
Let X be topological space: Two points x and are in the same path component of X if they are connected by Path in X (or equivalently if there exists Path- connected subspace A of X containing both of them): (a) Find an example of a topological space with two Path components and another with three Pa...
5 answers
Calculations Using RvcD 6100 ml"RVc (F)Drl(4.95/)-4.50 . Du T)~This 40O ml is the cstimated volume of Kas in the gustrointestinal tract
Calculations Using Rvc D 6 100 ml" RVc (F) Drl (4.95/ )-4.50 . Du T) ~This 40O ml is the cstimated volume of Kas in the gustrointestinal tract...

-- 0.022147--