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Here are the design parameters: You must submit density function p(x,y) , describing the density of = lamina on the square D = [0,1] x [0,1]. The total mass of the ...

Question

Here are the design parameters: You must submit density function p(x,y) , describing the density of = lamina on the square D = [0,1] x [0,1]. The total mass of the lamina must be exactly 1. The function must be positive everywhere on the square_ The function must be continuous, but otherwise can have any form you like, provided that the total mass and moment of inertia can be calculated unambiguously by the machine_ Your submission will be judged by the distance 6 between the two moments of iner

Here are the design parameters: You must submit density function p(x,y) , describing the density of = lamina on the square D = [0,1] x [0,1]. The total mass of the lamina must be exactly 1. The function must be positive everywhere on the square_ The function must be continuous, but otherwise can have any form you like, provided that the total mass and moment of inertia can be calculated unambiguously by the machine_ Your submission will be judged by the distance 6 between the two moments of inertia, defined as 6 = Iy' where: Ilzp6xy)aa and Jx"ecx,y)aa The density function p should maximize Ix, and minimize This is not meant in the rigorous sense here, but rather you should strive to make Ix as large as possible while simultaneously making ly as small as possible, so that 6 is as large as possible:



Answers

Use a computer algebra system to find the mass, center of mass, and moments of inertia of the lamina that occupies the region $D$ and has the given density function.
$$D=\left\{(x, y) | 0 \leqslant y \leqslant x e^{-x}, 0 \leqslant x \leqslant 2\right\} ; \quad \rho(x, y)=x^{2} y^{2}$$

Let me know. Let me know. Mhm constant density roll constant. She's intensity on your occupies This region, the X Y plane So x is between Yeah, zero on they Why I became so on age. So the Islam Inna Swift and we're looking up, right? Um, this region eso would like to think it the moments of generation I legs. Why on the the reduce off rotation this one about the, uh, with this off direction about the eggs access of these one satisfies the equation that is square. That was the total mass. Yeah. Yeah. Has to be equal toe you want with the veneration vision about X on the I would also like to find that George plant. They're already some rotation. Um, probably the exact is under this invitation about the Why, actually says authorities one or yes. Well, Barbie and you have this equation. So where this is the total mass? Yeah. Oh, Must, um Yeah. So for this one, hi, Bags is gonna be equal toe hold. This region is D. You're gonna be able to the angel. So the whom the so time, sir, or distance today y axis, which is gonna be wine. Well, that is square. Um, she would have the X. What? So we'll explode. Ranges from zero to be what on, uh, for our the movie and why it goes from zero up to age. Why we're still upgrade so well. First, neither of with respect. Yeah, well, you just x constant. So it's gonna be just be so our subjects would be, um, the reason first, he explained, Yet the I am so role, they go from zero up to age. Why Squared? Yeah. Why? I'll be minus zero on the all The drill of y squared is gonna be Well, the world comes. We have Why keep thirds s about following in there. Um, she would be What is wrong, Speed? Come, Sage Cuba Hoax is I bags now? Yeah, I said, why the into all over the same region. But, uh, you have thinking with X square, there is a distance today. We're axis. Well, exclude a round of the X. Be so they drove off. Texas court is gonna be so each of you. Thanks. The third power. But there were three that we can be zero. And that's why which is gonna be this will be, Yeah, be to serve power thirds times roll or that it's a constant we can talk about. They have the integral from zero up to age or B y it should be, um these are, uh, and slow them stage. So this is my so works. Otherwise, why? So these moments for inertia? Oh, no. We need to complete What is the total mass? Not that equation. The total mass is gonna be able to into Renta role inside of the religion that iss these also those into those, um, just be search does Asia. It is a rectangle of constant constant, basically. So the mass is gonna be photo damn stage. So we have this equation. Well, her for this value for finding his Yeah, that that number squared times the mass, which is thes number. I'm zero Beaton's. Each times wrote that should be equal toe. Yeah, I said thanks, everybody. Exactly. Which is a call to, uh, these number that she would do Role must be on stage Cube. Mhm birds. So well, we can waken lose again. You might buy Rosa Rose. You hear and see you roll. I'm h b zero. So because my dad, And then we can be very by the you divide by age. You have? No. So yeah, so? So they give us a That square is ableto um each word. Yeah. So both the the re radiates off Jewish and about the x axis is gonna be able to take this car up there so each very describe it off, right? And then all similarly, for this one you would have Yeah, that. Yeah, that that should be ableto to the moment of inertia about why physical toe be cubes. I'm Jerome. Search shirts to him. He goes all grow castle age. Really good advice. Eso We're gonna have that. The squad square? Yeah. Has to be ableto b squared third. So that leaves. Is that these? The situation is equal to B R. Disquiet off he So these are There is something duration on there. The one with the very show. He's whole, uh, about the x axis. Is this one and, um, Bogey y axis? These these number

Okay, So for this problem were asked to find the mass, um, radio duration of x and y as well as I want I i x So the first thing we need to dio in order to find, um, the duration factors was we need to find the mass, So it's gonna be from zero to be zero to h p D Y D. X. So this is going to be from zero to be and then p y from H 20 dx. So this is going to now be zero to be and then ph d. X. So now this is going to be P h X from B 20 So now it's going to be P h B. We're going to come back to this later. So the next thing I want to do is I want to find i X. So it's going to be zero to be zero to h, and then we want to do why squared p de y dx. So this is going to be zero to be and then one third y to the third p from H 20 the X So I'm gonna put the one third on the outside and then this is going to end up being from zero to be and then h the third p d x Then what I want to do is I want to do one third h to the third p x from B 20 So this is now going to be one third h to the third. Um, be PB sorry. We're also going to come back to this one. So now I want to find my nurse a based off of why? So it's gonna be from B t o. I'm sorry, h 20 and then beat Oh, or no, I did have it right the first time. So be an h. Okay, so then I want to dio x squared p d Y d x. So then from Peter. Oh, so this is going to be for D y It's going to be X squared p y from h 20 d x. So now I'm gonna have beat Oh, so it's gonna be X squared Ph. D. X. So the next I'm gonna integrate in terms of X. So this is now going to be one third x to the third p h from b o. So it's gonna be one third B to the third pH Okay, so now here. Here's what We're gonna combine everything. So are a different color. So I called. This is usually degeneration is symbology. It's going to be the square root of I EC I y over em. So I said I Why is one third? Um, let's see. I y is one third uh, well, CB to the third pH over our m, which is P h b. So right away. We should see that I have a h that cancels p that cancels. And then now one of my bees cancel. I can now take the be out, and I have the square root of one third square root of one is one over the square root of three. And then for my duration, Why, it's gonna be I X over, M Yeah. So I'm going to have one third h to the third PB over P H B. So again I can cancel out the bees and H is here, and now it's gonna become H to the second, so h comes out over square root of three

So in this problem, been trying to find information about the you're trying to find moments of inertia about X. Why anti origin? Also trying to find the radio operation about also just three same points X axis. Why? Access and the origin. So what is her lamb? You know, Should be the first thing he s every loving A is, uh, rectangle. The murders is at 00 60 03 and six. Common three. So it's a rectangle, and it says he's a rectangle and make some limits easier. We're gonna see how so. That's did everywhere to seize. Now let's get started. Let's find the moments of inertia. What to do that I need to first. How did density that's it arose in this question is guru. That's come on. What x times y I mean screwed up ex wife. All right, so what is the moment of your show about Oh X, that is defined by the double in a girl off y squared? Um, Row D A. That's a general formal. And now let's find it for this case. So we have to double in people. Ah, why scored row and row is extra 1/2 times y to the one giving us to double Negro off. Excellent. Would have times y to the five house. Now I need to find my d. A idea is just X de y. Now, what are my limits? Since my since I have a rectangle, I can just see the limits easily. X goes from zero to sex on. Why goes from 0 to 3? All right, now, let's get to integrating. If integrate this first, taking her in a crawl will get the interval from 0 to 3, uh, to third X to the, uh, three house. Time is why to the five house Dr Evaluated from 0 to 6 on the whole thing d Why, All right, so now let's plug in my six. I will get the double the interval from 0 to 3. Now let's take out the 2/3 like in the sex. I get six rats, six for excellent story house. Now, why to the five house the why? Well, let's integrate dot Again, I'll get 2/3 times six right, six times. Um, why, to the seven halves evaluative from 0 to 3 now also have to seventh out here because I integrated the five houses seven house to over seven. I'm just kind of multiply them right now or a simplification, and I just let's see what is 2/3 down to seventh that is four over to anyone. So for over 21 next US plug in my 30 doesn't matter because it's just a zero. But the three will give us for over to anyone. Times six, right, six times three squared, uh, issues was gonna become three square times and I times another ride three and then another three idea take increased certain white to a seven house, which is why to third plus of 1/2. So why, to the third gives us a 27 in total and the 1/2 years US Arad three. They gave 27 right? Three. Giving us a total of nine can make some simplification. 21 becomes a seven on my six becomes a to I want to play holding numbers. I will get AIDS times of 27 times, right, six on draft three. Which gives me if the reality outside rap too. Now multiply all of these numbers out. You should get 648 Brad to over seven. Now, that is the inertia about X. What is the inertia about? Why the initial about why is Well, um you need to adjust to the same exact thing. But this time we need to instead of having double integral Oh, Weiss girl, uh, grow. We need to have the double integral exclude row me a Now we see that this is so similar to what? We have people decisions cynical from 0 to 3. And then you go from 0 to 6. Oh, now, this time, Extra five house. That was why? To the what? How the X do y Meaning? I don't need to integrate it again. I know what's gonna happen. Last time I ended up a bit of 2/3 times a 27 times as three to the seven house Tom's a six to the the rehabs. But this time I'm just going to It's just 600 the week because I'm solution. I'm integrating forward the order Prior variable giving me this. So now I get this huge number. And if I want to play it all about all good 2592 right to over seven. And that's why in your show about why you just need to do the math to get that number. But what is my inertia about zero? So my inner show about the origin, it's justice. Some of my 200 shows, which is they could come into. No, me under seven, Right? Two times 648 close. 2592 giving me a total. I have it right here. 3000 240. Wrap to over at seven. Okay, great. So now I got all my moments of in your shelves. What is my radio? So, for that, I need to find my mass. Must is a double integral over all the A, which is so simple. Don't you go from 0 to 3 and then you go from 0 to 6 of Exeter. 1/2 times why you don't want have the eggs, do you? I now this is similar to before, but instead of having 2/3 Time's up to seventh, I have 2/3 times to third because I don't want to buy anything by ex skirt or y squared. Now I'll do here. Three wrath. Three times six Rat six. Because I'm gonna do powers of three house for both. So now with this, I will get this three castles with history, just the re castles with sex giving me a to so I will have eight times wrapped the returns. Rat sex is just 24. Ride to Okay, great. So now we have the mass. We are already to find all the radio. So the radius about X is the square root Oman inertia about X over a mess. It's just this crude out 648 right to over seven. All of that bull were 24 right? Two rats who canceled back to castles 648 divided by 24. That is going to give you 27. So this council there's 27. Therefore, I will have a three wrapped three over around seven. Sweet. Now what is my radius about? Why? But that is the chemical at the screwed off when inertia about X over months. So that gives me, um, 2592 breath to over seven. 00 are Tony for right to all under this parable? And though I castle disc ERT wrapped tooth in $2592 by 24 will give me wonderful eight 108 So what is one of eight? 18 if I factor out? Three is a 36 so I will get 36 corroded. It's six that I have six frat three over. Arad. So nice. What is my radius about the origin while Dad, it's just the square root of radius about X squared. Close the radius of our y squared. So for that, I can just factor at the rest of you were at seven because there's gonna appear is in both of. And then I'll have to screw of nine both 35 36 giving me a 45 which is factoring out of three having a five left the rear grad 15 over around seven. So now these are Are are all my answers in your show Biotics You cost too. Um, 648 rat too. Over seven. You sure about Why is 2592 breath to over seven and inertia about the origin is 3200 for direct to over seven radius about X is three rad three over at seven radius about why is six rad three over right? Seven and radius about the origin. Does the re drive 15 or around seven and I'm done?

Pushing me to find the sister of math, Philomena. And what is ever Lamanno, huh? Her love, you know, is gonna turn out to be 1/4 of any lips. So we're told it's any lips The red one and the Greenlee region on a blue region on the intersection of call over, which is this quarter? The Libs in the first quadrant. We need to find a central mess. To do that, we need to find the ex. According it The Y Coordinate it. The X coordinate is defined by the Momenta. Why? Over maths? A momenta. Why is on quoting until why he's a moment of X over mess. I had to find this information when you to know what the offensive were given 10 cities this crudo nine x skirt lost twice foot. And now we need to find those information first. Need to find out a mess, which is the double and roll off our density d a. So, plus fine ever math your last Use the calculators who put in our density equation. You put our eggs on white variables. The limits from why is from zero 21 the limit up x on the other hand is from zero to the side of the curve. What is that curve equation? Well, our the lips has an occasional nine eggs for plus y squared it mostly want to. Let's isolate X. I get X skirt is one minus Weitz for it. Overnight, the sector school roots are gonna be left XP calls to positive. Negative, Brad. One one minus y squared all over three. That sense we look at the positive end, you can make it just hot fuss. Not a minus. OK, so that's our limit. And every computer that we get high over 18 down here, that is, Arum. Eyes so mass because to hi, over 18. Now what is my moments now? Well, my mornings, my moment of eggs is the same as mass. But instead I'm gonna integrate. Why? Times wrote the eight. So let's go to hear Ellis shows. Uh uh a why To the outside of a recreation. So what is at a moment of eggs? Our moment? Allex is it's a loading one over told. All right, so moment legs, he goes to 1/12. Now, what is my moment? Why? What are what you similar to moment of eggs when you said Why will have an X Road d a. So let's go back to her calculator and let's make this why an ex and so want over 36 which makes absolute sense. So Mormon told, Why is long over 36? All right, so let's find our X and y coordinates. X coordinate is momento watched 1/36 over a month. Hi, Hi. Over 18. So simplify this becomes a to on my ex Gordon is wrong Forward too high. Next let's find out why coordinate It's the moment of eggs one over tool over mask high over 18 simplifying You get three and two and so do we Over to Hi is my light cordage So I have co ordinates 1/2 pi Come on, three over to buy I mean to crafted So let's craft Our center first is I'm delighted by two Hi, Mary. How three divided by two by But let me hide the cube or And this right here is our central mass Eso we're done


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