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The moment of inertia about the origin for a lamina in the shape of' region R is given by the integral JS,(x2 y2) p(x,y) dA_ where p(x,Y) is the density functi...

Question

The moment of inertia about the origin for a lamina in the shape of' region R is given by the integral JS,(x2 y2) p(x,y) dA_ where p(x,Y) is the density function. Set up polar double integral t0 find thc moment of inertia for the part ofthe ring in the first and second quadrant between the lines and v3r and the circles 16 and r 100 where the density function is given by plx,y) = (x+y2)'_ Do not evaluate. (10 points)

The moment of inertia about the origin for a lamina in the shape of' region R is given by the integral JS,(x2 y2) p(x,y) dA_ where p(x,Y) is the density function. Set up polar double integral t0 find thc moment of inertia for the part ofthe ring in the first and second quadrant between the lines and v3r and the circles 16 and r 100 where the density function is given by plx,y) = (x+y2)'_ Do not evaluate. (10 points)



Answers

A lamina with constant density $ \rho (x, y) = \rho $ occupies the given region. Find the moments of inertia $ I_x $ and $ I_y $ and the radii of gyration $ \bar{\bar x} $ and $ \bar{\bar y} $

The part of the disk $ x^2 + y^2 \le a^2 $ in the first quadrant

We have this circular. La Nina. That is really the first time quadrant in the X Y plane is, um, Circle Quarter of a circle of radius A. So we have here X on the y axis his piece. Um, this region, the that is our la Nina has satisfies the following equation X squared. Plus why square so there's than a square on is in the first off that first often. So, uh, first quarter. So, uh, that means that eggs. Why both? Positive. So this piece on the world has constant density. Yeah, and you roll and that we would like to find the development of the nation about eggs, about white on the really off Jewish in these numbers that these numbers satisfied equation. Oh, well, I don't buy the square this this number squared and it's the month. Uh, it's gonna be equal to the moment of inertia about X. Um um is it was the month squared. She goes to the moment of generation About why? So now for these, Well, let's first find the mask. So, since it is a consultancy, think the mass is gonna be roll. Thanks the area, but the area of this is the fourth. Uh huh. Off a circle over reduced eight. 1/4 off a square. Despite so that is a much And then there, all the moment of generation about the X axis is gonna be people to integrate inside of the Oh, yeah, White squared because the density you ate so y squared, uh, in polar coordinates. Physical thio are has signed forever is why our square will be art square. Sign off, get out squared and then differential of area is young, are are different People are coordinates Yeah, on this region is described violating our go from show up to A on our on Google's from show up to my house But it's this angle our house So CEO by perhaps for White House for freedom Well, we have the first thing to go for Cube Yard, which is our fourth, where by force with dive always okay and put so well and we have diversity. It's gonna be 84th. The very best for minus zero is okay before about your life, for that was the best you can. Then we have this thing to go. That internal left internal from zero up by House. 40 off sign squared. So this crew, huh? Uh, yeah. Yeah. So Science square is single toe. There's a general educate I need that tells us that this is going toe one minus school. Sign off two times, terra Right to so the central talking to know on the this would be ableto their hubs and then minus Ah, my sign. Oh, Pennst era divided by four. So they will be that following the heating by house zero eso is with the equal toe. So perhaps the 5 to 05 faults minus kind of depends by half. So sign off by about four there, that zero is gonna be zero minus. Sign off. Zero. So sign up. Zero is, uh this number is Syria sign of fight in social zero so that these, uh, business by over four. And so this moment, the finger she's gonna be, well, this number inspired before. Yeah, so that the moment he knows it would be that waas Yeah. Yeah. Um this is someone definition about X and for the moment of inertia. About why? Yeah, uh, you do the interval inside of the reason that is or in a what a recording. It is gonna be from sheer played, I'm sure upto I have something differential various d r RDR And so we'll excess our call sign. You need to enter with these. That was the density on wall that square. This is thes parties X square because the moment of inertia about why would be grow a list with the moment operation inside of the region. Wow. So, by doing this well, we again integrate. No, cube, you are under invaluable awaiting Depends on a show it's gonna give us. Wrote times 8/4 power. You are about four, and then you have this integral. It went from zero up to buy hats off course I'm square. Screw fitting. Yes, your style with, uh, Hussein Square a fitter, um, before So here. Um Oh, there is a similar identity either. Okay, because Sinus square that tells us that science playful angle We spoke to one plus because I know two times the angle. If I would buy two eso Well, we went away. These? Yeah, we're gonna obtain no your house and then plus oh, sign well off throw terms. I'm staying up like my four. I'm staying on divided by four on the well. The bouncer the same as before. Yeah, became perhaps and zero on in the similar way as the bounds canceled. Here they will cancel also here because they only differ by a sign. Here we have a minus that we have a plus, but zero is ableto minus zero. So this part who had nothing at you I'm Dana Internal is interview the same Just I have the four vital. So that is five hearts. So the same. Interval us in there on the way we would end up with these two people. Brought obviously your foot. You better, right before times five. Yeah. Oh, so this is I said, Why so well, as you can notice here, physical toe, the moment, the veneration about eggs. Mhm. So like these mhm on since the ready off duration. Oh, this is is related. Thio. I survived in this way. I'm easy to our e said Why is about to buy selects then? These two numbers will be also the same, so we only need to find one. So well, let's zoom this one here. So we have but number squared times the mass, the masses row time. Spy them a square. Anybody before that has to be equal. Thio. Mhm These things more entrepreneurship Role was 84 from spy. You know, like for sports eso. What we can do here is we can cancel one. Or do you? You can divide by a square. Cancel that And then we did right by you square Here we get, uh, power off to there and then we can cancel the density on by so that we well, his waas that number's square So that, like taking the square root, what would be left here is, uh so this number would be square root. Okay, squared by the wind Four, which is a joke. Mhm. So this is the the radio of duration and then this number is equal to Yeah, So for this happens because there is symmetry with respect to the sexes. His access eso with shape is symmetric. We swap x by y and y i x so that these two are equal on those to quite this far equal from the thes numbers that is the most version this already radio of Jewish

On the phone ownership. Um, is this card made a girl sign of fix? Um, between Yeah, sure. Under exposed by. So I have this curve. What a world that is between grab buries for X beating on this is the X y exist. Why? Because you're on five. Yeah, but the Islamic constant thanks to the row. Yeah, this is sign of tricks. White this'll, um, in a d and you would like to compete. The moment of generation will be exact lives if they rotate, but around the x axis, that would be like dating that night. Well there or immunity about the y axis. What would be e said? Why the corresponding Rabie off durations of the is satisfied? Uh huh. Right. Reduce obscuration about the x axis is going to be This is people to end. What forms A I x men locked. I'm a Louisiana. The other one, he was laid off. Uh, this is And what, um, in ice? Why? The whole body and some should be ableto one of the prisoners about the eggs. I see something Gonna lose them. This relationship. Well, um, like to find first, what is him? Everything. What is the total mass. So the total one is gonna be people don't integrate over the vision. So this would be thank you from zero to sign off eggs. That is for why on the bar for those are the bonds for why? From Europe. The sound effects on text was from here. Good bye. So the integral of the way is just the why, Jehovah? What? Evaluated A sign infections You so don't belong, bro Inter from the occupy off sign off X fix So the They're they're 12. Sinus minus. Hussein Should have minus cause side effects. Not waking up by monsieur so minus programs because signed off by minus persona. Yeah, represented by his mind is one Cassandra is one. So the review wrote times to to roll. That is the They must Yeah. Now, um, the moment of finish about the exactly This is the jewel they were inside off. Um, why squared? Uh so Well, don't continue it. These, um So if you have one back from Syria, some of fix and then or X from. So why So? Control of Rose Square is sometimes density since it is constantly Don't pull that out. You don't were over one squared. He's why, Becks? To have a third and then sign of fakes. Okay, Cube. The eggs became too five. Well, sort of physical tour. Same square. Yeah. Thank sign. Thanks. Let's go here. Designed square Come leaving us one minus course word so that we do a new substitution. You go. These so that Do you use cable Thio Hussein off from effects. Then the new is gonna people toe minus son affects the next. So the idea would be what? That humble look with the substitution minor the Yeah, so the draw off This will be minus you, then minus minus. So plus you the third power. Like See, um So what would you be in co sign? So by evaluated pile on zero wall. So this would be so for our view. Therefore, casino fire that is minus one. Uncle CYO, see everybody. So what? Oh, cool. So be like these physical thio minus minus one. So one No, there is one. Yes, sir. This man is one thirds on that minus. No one was one. Yeah, All right. Plus one third. So these stopping the signs is equal to both times one minus 12 No one minus 1. 30 people toe. He sucks. Third, someone would be four points. Um, she is not on this number. Mhm. With the four thirds so happy for thirds from some road, you have Ah, third there. So like this, we're talking about the X axis. Now, um, the moment is about the y axis will be mhm because we know X is quick. So nice squared. What do you wife, bull. Thanks, but we concealed inside a fix. So you I'm fine. And then we have diversity. So in general, he wanted his wife. It's gonna be row was intra from zero off the pie X squared times. Okay, sign. Thanks. Yes, nothing for this thing to go. The camera. You can make out which parts to the global FX squared Scientifics? Uh, yeah, was the goal. Needs to be, uh, so be you So that the musical to deals if the usable minus sign affects the extent you would be called to Yeah, uh, minus sign. Perfect. So that the musical toe sign affect the eggs and then Or the 12 this is in relation by parts Would be, uh, you have a nexus square here. Excess for she would be Yeah, by integration by parts is gonna be minus X squared and sign and then minus minus. So mhm when the DNA Oh, internal the the you support to new times being minus the u the Bangles You d v So we differentiate who works. You have excess. Quite. You have two X So is gonna be with minus and becomes a plus. Plus uncle side of fix times works. Yes. And then we'll this integral Yeah, you're in general is able to minus X squared. Placental fix. Plus that integral on being for this interval, we can do again. Integration by parts I'm here. Okay, which is cousin affects the X to be our the union this case? Uh huh. Yeah. So if course in affect the excess, do you then you will be able to rules for this other integration by Fox Do you is gonna be side effects the X sort of X. So that view is ableto sign. Thanks. So then these interval, Mr Billy Joel B minus X squared side effects and then sign affects plus sign off checks. Uh, force Twix amandine minus differentiating this just get to the end. Times sine effects ex. My major off off sign. Internal of minus sign is, uh, assigned so big. Three times to sign Catholics. Good. So these works Sign affects minus X squared for side effects S So this is the entity ready for X square? Sorry. Clicks. So that way, evaluate between zero on pie. It is your own pie is gonna be ableto these evaluated between, uh, by zero. So we're gonna have minus five squared, because I know over by is minus one on the sign off by zero plus two times, uh, personal fight is minus one. Okay, on them. Minus. So these at zero would be zero squared. Uh, plus Syria does that. Then plus two question of zero, which is one. So this would be a minus two. So this is gonna be minus pi squared times minus. So glass pi squared. Minus two minus two or minus four. Um, so this interval is, uh, by square minus four. This is spy squared minus fork. All that is the moment of inertia. About, uh, there were Axis is gonna be alright. Wrote them by squared minus four. Until we have this is my legs e Why, then we'll, uh, or must or mass waas equal Thio two times roll. So then we have to have the following equations. We have let this square times, uh, the mask that is to guru. He comes through that students wrote that should be able to e bets that is equal to, um all roll over three square or roll over two squared. So for canceling row, that is no zero then behind anybody to here. So we divide by two. You're gonna have divided down to cancel. Want to there Thio have that these number taking the square root? Uh, well, it's gonna be a screen off to over three squared you goto square off to you got, like, three. And then for this one, you know that that is squared. And two times throw should be for two to my somewhat. That is equal toe role. I'm Spy Square minus four. Okay, uh, this is, uh, from the finish about the y axis so we can cancel this. We can divide by two. Okay. Set up. We have, uh, on this, uh, reduce off duration physical toe square root off by squared minus four. On that. You have those through novels? All right. I'm not in the moments of inertia. Oh, What

To find the boundary for by X belongs to -1. Mhm. You do it X squared equal strips plus two. Some place X squared minus explaining to equal to zero. Yeah the bountiful my between the two given curves and the mars becomes and equals to integration -1- two. Integration X squared takes this to K. X. Squared beaver. Big not solving the riddle. They get integration -1-2. Okay excuse -2 x squared minus key export for big. Which can be simplified and we'll get -177 appointment ticket. Next we find a mix and endless So M. X. equals to integration -1-2. Integration X squared express to the X squared back be very dear. Now solving the inner interval be good. Yeah Integration I guess one good too exploring food. Let's vote. Excuse just 40 square minus explosives upon. To what if we go to 531 17 people. Okay. No girl bloating and I'm very equal to Mhm. Integration -1-2. Integration X. Squared to express to K. X. Cubed. By solving the inner and middle. It will become Okay. Integration -1-2. Explore for just to excuse minus explore face the and it's equal to 18 Upon for you. Yeah. China's defender of the mosque mm minus excessive in coma minus the individual Upon 1 77. And the mosque is and the masses -177 upon 20 K.


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