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Centroid Find the centroid of the portion of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ that lies in the first octant....

Question

Centroid Find the centroid of the portion of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ that lies in the first octant.

Centroid Find the centroid of the portion of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ that lies in the first octant.



Answers

Find the centroid of the region bounded by the given curves.

$ y = x^2 $ , $ x = y^2 $

In this problem, it's required toe. Find this in Troy, which is the center of mass off a region off off a circle X squared plus y squared equals is quick. So he's asking Boat these circle with X X and Y X in the first quadrant. So he's asking about the tree Asian with article two way radio equal to it. We can start our solution boy Completing M as integration from zero qui and integration from zero to square root. It's quid minus X squared. Do you, Roy the ex You Roy dx. So it will be equal toe square root from zero twee four square root is Quito minus X square. The ex which will be equal toe. Why a square over four and we now can complete also m floy decoration from the unit we fame, integration and fame Boundaries from zero to square root squared minus ex squeeze eggs. Do you want the X? It should be equal to integration from zero to we thanks square root a square minus x square The ex equal tow cable 3/3. Now easily from the cemetery, we can calculate export which also will be equal to why you bore from symmetry equal to M y over m equal to four e over three. Boy. Thank you.

So you know these disillusion? These, uh, bounded by the planes? Yeah, and by why Call zero explain c equals zero in the plain. X equals zero x Y No X. See Runda. Why are you here? Why? No. You have this region here in the first stop tint when he saw Zoom are lying inside eliciting there It has, uh, c equal to the radius. Rhonda this gone, gone, gone, gone And he said, I'll be sitting there. We're used to to the Ciccone. Miss Part of their that is a cone c equals two are were excluded Affects squared plus y squared on the sitting there, through these in Sami's I love Oh, these outside our It's even very saw X squared plus y squared it was all for Well, there's some water over here. He said, Do you want to find the sentry? We'll be this region. First we're gonna find what is the volume? What? So we do cylindrical. So, uh, we're angle fitter. Oh, sir. From zero. Who? The house. You don't see out of my house. Shear off. But a house. Oh, are those from Syria off too? Two, please. R squared was a morsel. Listen, you square syrup to do our Did you think it over? You are. You see egos from zero off to our ma zero are on the volume element, sir. So that way we degrade first This part he said, Uh, you know the truth. Bye, House. Perhaps this angel from zero to my house. I'm not the universe. Isn't there so good? The reading z until easy. That's just see well with the 18 0 on our use. Ah, so these uses turned down into the drug Things you do are getting our next two are so artoms. Your are from these internal TC You are square. We are said I'm into all these are cute Uh, our Cuba nerds Thatta boy routine. Do you want to hear what you have to Cuba? Thirds. So that thes volume we'll be able to buy times drew you perhaps through Q thirds sort of its councils? No, Why? I'm still work. Thirds volume. So what is enjoyed? We'll use thes volume, but our well, by the symmetry, respect to theta this android has do How very few Gaza coordinates. Uh, Seon Choi. Uh so the coordinates are well there. See the Jungle monster bitch. Well, um she's a distribution. Uh, would respect tool. There is. You seen that trick? With respect to the center, we should be the bangle destroyed That I go There is Morrow by four. What sort of this intro you love you sound a little wild boy for for G. We need to find ours. See? Just gonna go to we do it Well there. We're good volumes here. To my house. Yeah, No. Are those from zoo Two? Two One? See, It was from ah c O r. And then for diversity multiplied by c years ago with the Red Sea over there, will you? She's done so when we're integrating internal CVC is equal to C squared Popes. It's a disease tolerate between are it's you? Yes, r squared hose of the from bass intro. And then we can also integrate Sierra. That's years we're gonna be Yes, my house. That was that. There's a draught from Syria of two over. You are thems are square house. We are so in doing our cube which hasn't be there. They've gone to the four by four. And so that evaluated between Seo Until we go to. So you have buy off your true squared. Thanks. Uh, two of the four hour you want in life. Four. Uh, So these well, today for us, with the discourse, we're gonna have to before and I would like to do for which is going to what? So the average C is that we'll divide it by the volume will be and they removed the is, uh, said these parties by to the reverse disease by undergoing off the she wanted to buy over for Well, bye. Well, before you were three is equal to multiplying down way down. So do you want to tree? Why over four? Why? So this is the hours easy. His army three four, and then they're ours are We're done. We need to do well. Being great over are being the SAR comes from the volume element are you know, so see goes from zero up our all right It was from Joe Toe thing that goes from zero off dude Bye house seem to really first respect that are out. Well, he hopes there's that, uh, holding drill over. You see, it's just see on deniability on deer in art is gonna give us army are we have our And there is some risk orders is love. Yeah, our cube penalties by house on. Then these are Is there our, um that is entering from zero up to to our and so will the girl over our Cuban visa noted by house there, Um, a girl of our cue bees aren't to the fourth power y four. Well, I didn't make it zero to Jana Mosa bhai them. So, uh, to this world power I like ju damn sir for that. He should do word so that that all these stupid or four people toe so they were legal to buy before to third bars is gonna give us to buy undying well when you don't divide. I didn't volume so that the average are is simple to buy on. The volume is equal to for by with three buddy buddy four by over three people is to buy is going to buy only one. Too bad that Did you have me? Trainee has to buy over. Um for all right said that councils that Andi, these four is a two times two. So how long That is sweet booking three house this is the hours are and so that, uh, these really three paths. So this enjoyed will have coordinates. Three hives. My fourth, 3/4. So those already are better. They're in using that on the C cording. So that is this in Troy? Huh? This in front of these do?

Well, let's hear. Uh, so it's here. Here, X Y Z on Duh. Sean is bounded by maybe. Uh well, they're already sees Row on their way here. Very lies. Or the school this ponies go to There he goes from zero to buy for. You have these, uh, this region. I'm boys inside off this pier. Roll with a J. It was very cool. Very good for needs. Very nice region. So, um, I like to find this in Troy. Well, so what is? You have to find a way. Six. What are expire? We learn CPR. So, uh, by Senator, you can see that all this region is like you have these Mr. Wurtzel cut off that that stuff, but ah, we're working. Uh, while I'm being irritated. Our home Vera. So they're a little He's my symmetry. These have to be you. So, uh, you only need to compute ways up are so first let's cook it. What is the volume off this region B. You will You will be, you know, the equal to from you too. Bye. Nice for a fee. Our egos ground 0.44 Ana Robles from syrup, eh? She's inside of that syrup doing row now The morning relevant role Sign off E. She meant a great row. Be waiting row, squared row. But I still do No que birds So we have a week there. So, um, they you thirds, you can also be seen trouble We respect. Oh, should, uh, they dropped by. We've got several issues. Nothing there. Depends on thermals are usable just by now that I'm stupid. I being at the scene to go left from zero up, but over four. Oh. Ah, Sign Wolfie this fee. So the drill sign in your old sign. Me? I know that it's mindless cosigner. Um, so that blew it in between 54 on DDE zero would be just minus and in. I was saying, before miners zero so close. I know whatever for is one of her squared off too. Cassandra zero is one, but this one can be reading us squid off to who was going to sort of these numbers? Miners? One. Oh, Linus, This group of cool very school of I mean, who began flip this obstruction to give Really minus. So be still, volume is through by times. A cue four words that was these number. Ah, the order flipped school of two minus one over skirt to resemble you. So this is the boy to know for. We're what is, uh, see bar you have to do. When did you get over the region? See, that is going to roll them school. Sign off he Bye, zee. Oh, Ro goes from zero up three and we have the volume element. Rose Square. Sidle he It's, uh Well, this is the role. And then Feagles from joke by over four, the fee there goes all the way around zero off to buy. So, uh, this is Sophie. We're integrating. Firstly, doing Pro Cube. We do it roll. I'm homeschooled. That's real cute. These people go grow to the fourth power divided by four. So you get that after a on dina. Also, they drilled from Yeah, never these central zero to buy a sequel to just two. But suit off. These would be equal to to buy times. These are a my nose 0.0. This is gonna be a to the full of power. You better wait for minus zero. So just that I'm being you have the surgical left Betrayal with respect to tree trump zero by over four Cassandra Fee Sign Alfie D feet. So home you can do in your substitution we call sign of fee to be you. Then the U is equal to co sign a free the fee. All right. Said I didn't have the central of the internal off. You. You. Ah, So they give you two. You squared hopes. So ya like these became zero by four. He's gonna be ah, use squares off. Use of the signs of sign off by over four squared. You bite too. Minus sign zero. Where do so 000 signed off by over 4 December. You say it's one of ours. Good. It'll do so that this court, these, uh, this whole thing year we'll be just, uh, one. He's a squared. That was 1/2. So you want one? Have 1/2 more. Now squared, Dina. Well, that is for got bored. And in that times two. Why time's a two fours forward everywhere floor. These concepts went over there so we wouldn't get very with people too. Um, bye. Time's a over for you might have about eight. So this is all for today, Steve, are we need to divide by? The volume would be so save our would be these integrated value off. See? Then why were you so mean to read it by you? See these? I'm David. If we divided anymore, you we would get you need to divide is number by No the volume sequel to So we need to divide my huh? To my Why is it you? Why don't you third move in terms? You too. This one over square. So, uh, cigar is not viewed. These number couldn't see viable people toe these terms. That was that over that comes up. So we're people too. Um, bye them see Fourth flower. There's three. Uh, well, we have also I scraped off two on the top. It's a squirt of two divided by Yeah, uh, them screwed by you Agree? Third hour. Those for the school minus one. So, uh, no, you can't cancel. Is three all the bars off They are gone because the council bye. So this whole thing would be so seedy. Bar is gonna be simply final that we have her three 3 a.m. C. Squared off. Cool. 3 a.m. C. squared off, too, over this number 16 $16 script of cool miners. What So, Doc? But he's gonna be Steve are on bass intro. Another score. It's expire. Why are? Because you are so these other two are zero. I'm Sierra one. She wears that. That is this enjoyed for these?

So you have of this region. Um, but, uh, in the well, I know that them you hide, you call the c girls like, uh, this Linus Well, r equals two. See, because they're scared of fire where bizarre is them. Is it the sense that I am the access that does not like these one on the way to the exit plane? And then we're taking down around like that. So you have these miss figure on duh. Be so so inside of the sea in there. Overuse for you're assuming they're here. Well, is that all you see there? I'm being Yeah, it is this region, so Well sees me, Dina. Zero on the square with the flowers. Rhonda lies homicidal. This vision she would like to find this century. Sandra Dee. Where is that? Yeah, so by the secretary. So this American respectable. Yeah, Mom. Well, the never is next. Um, Why there's you Well, the the distribution off, Marcie. Symmetrical respect through the origin. So? So we need to find a leading to find yourself. What is? See bar. Hold on. Let's compute. What is it, boy? Divers. We're gonna do cylindrical according it's the so here they have also the way around by. Yeah, well, are those, uh I'm zero before syrup before our I don't see those from zero to It's quoted before, so But then the volume element is our cylindrical. So we do praising Tolo. See, you have a museum was drill. See you the drill. A breezy, you see, so evaluated in zero I was group a plier. We was just screwed the fire around. Um, sort of more We can do it because nothing here depends on terror Will just get through it on Dean Puree. And that would be intern from zero by Thames eater from Europe before our times These comes from the sea in diversion struggle t screw that square root are aren't scared of fire That is equal to are the three hubs So the view degree that we will get him are today five house. Oh, well, it's like that power increased by one and then you have helped me Bye bye ingress of that number to divide grams through over fight so not done these number the boy will be to buy times who we're five times for. Hey, the power squared of that survival people. This number is our culture. 22 54. Because taking the square room for his too. So, uh, so that our whole the volume was gonna be Oh, you have. Ah. Why to today why I'm being another another 2 to 37 power. Damn spy united. Lifelike. That is no value. So now for Escobar, you need to do you to divide by volume and then we integrate where the region, Uh, nobody will see you have the the Jacoby Undying. The volume element is, uh, these part. Are you see the yard? Yeah. Two heroes or seen bones to objectify. I was from syrup for unsee goes from sear off the squares of our the same bombs. That's before the well, Now we have a 20 year old were see we're doing Z. They see the trail of that would be, huh? See c square house, that is. Well, we could be thine. It's good to fire. Zero would be Screw the fire word like do my zero. It's a daddy Suggest our house terms No, are. That would give us so. Our house. I'm sorry. Easel R squared house. You are, And then we can be in trouble, mister. He's gonna give us a photo to buy. We're getting off this in general Strong zero before. So the central where we can cancel that two timing the school. You see a girl he's gonna be well are to the third power. No, very sweet. Our third. They're about three on that computer between or zero the vehicle to my two Roman rule. So vehicle, too. For the third power. Wait three. So that these part or C bar you gonna be Bye thems for anybody. Lacey. Remember Boulay move in. Sequel to why them spooked the serum. You don't buy it. Them screw seven. Well, so these, uh, he's not very physical, too blue squared. Our people do, too. What do we got? 376 26 says not be Well, this division is a courtroom. Bees One time Southern, they would buy times for two for the six. Mom, I'm slide divided white Nissan themselves by well, stems to seven times three. So that, uh oh, but six councils, all the bars only yet just want overtime dreamed up by by. So So that's Ivar Should be too Live over. Dude, I'm 3/6. That is so far. Rhonda. Yeah. So explore. You see 10 Why Bars? People do zero on Steve. I received 5/6.


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