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Oima constant-donsly Paia ahotn in tha (iguroright Acaume density 0l 1-Fnd a mestend Cenior(1D,01Tn mer # me(TypeIeacransrsmhiedhoml)i double Iniagral0ives Mythe plate s frst momant about the Y4n Uco incroasing lintits = TintegraticnJo(Type axuct anaten )Betup Ma goubla inlogral inat @ves Mlz , the plare '$ fir3t moment about ine >-4553, Use Increasing lintis integration;(Tyou cxact ansmon )Tna conter ol Inued locaied "D (Typa - odelod pae Theu a Hmennd uimple#dtorn )lente nouangwr

oima constant-donsly Paia ahotn in tha (iguro right Acaume density 0l 1- Fnd a mestend Cenior (1D,01 Tn mer # me (Type Ieacransr smhiedhoml) i double Iniagral 0ives Mythe plate s frst momant about the Y4n Uco incroasing lintits = Tintegraticn Jo (Type axuct anaten ) Betup Ma goubla inlogral inat @ves Mlz , the plare '$ fir3t moment about ine >-4553, Use Increasing lintis integration; (Tyou cxact ansmon ) Tna conter ol Inued locaied "D (Typa - odelod pae Theu a Hmennd uimple#dtorn ) lente nouangwr Leach oltne answer Dakus



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A plate occupying the region $0 \leq x \leq 2,0 \leq y \leq 3$ has density $\delta=5 \mathrm{gm} / \mathrm{cm}^{2} .$ Set up two integrals giving the mass of the plate, one corresponding to strips in the $x$ -direction and one corresponding to strips in the $y-$ direction.

To find delta. For the face. But we have one divided by two to be equal to M. And M. It's equal to our movement is equal to from 0 to 1. The W. Integer. You have Y squared to to Y. My nets Y squared. You have your doubts sir. The eggs do I. And this is equal to This is equal to two. The author. You have insignia From 0 to 1. I have y minus Y squared the way. And this is equal to I have to yeah it's a Y squared divided by two minus Y que divided by three. The interval is from 0 to 1. And this gives us mhm. The pizza one divided by thanks. So this implies that My data it's equal to three divided by two Then the 2nd part B. To find the average value. So our average value let's say our average average value. It's equal to Your integral from 0 to 1. You have from white squared too. Two minus wife's great to Y -Y is great. I have white flats. one the X. The way Divided by names Igra from 0 to 1. You have white sprayed two two White mine. It's widespread the Y the X. The X. Y. Rather you have the X. The way and this is equal to. So this gives us one divided by it. Soon. Then the denominator gives us one divided by three. So this thing is going to be equal to three divided by two, which is equal today. Thank you. So this implies that the values the values are the same at the same.

Okay, so let's make a picture. This is our curve. This is the plate. The plate is sandwich in, sandwiched in between X axis and the curve, by definition, off the center of mass. Ah, the radios Victor off the center of mass equals, um area integral over the plate over the plate off the density off the plate multiplied by the radius vector X Y and multiplied by the area element the x d y. And, um, the bottom of the fraction is just the mask off the plate. But the mass off the plate can be calculated as a double integral over the plate off just the density multiplied by the area element. Um, Now, because, um, the plate, um, is normal. With respect, Thio the x axis, we can replace this double integral. It is to, um, regular into growth. The external integral is with respect to X X varies from 1 to 2 according to the problem. And the internal integral is going to be with respect to why and the internal integral is going from zero to F at X. But FSX according to the problem, um, the equation of the curve is just too over X squared. Um, sorry, I didn't I didn't draw the other side of the curve, but it doesn't really matter. Okay, so this is from 0 to 2 over X squared X y x y. With respect to why, um, the bottom of the fraction can be replaced. Uh, these, um, to regular, Integral in a similar situation. Two over X squared times the density. Ah, times de y. Now, I remind you that the density was just x squared. This is just excess square. Ah, let's any right. The top off the fraction So x squared, multiplied by X gives us x cubed, right x squared, multiplied by Why gives us just x squared? Why, Yeah, well integrated with respect to G Y. Mm. This is just X squared, integrated with respect to the y. Okay, X cube. Integrated disrespect. Why is just X cubed way? Um, X squared. Why integrated respect? Why is just one half x squared white squared and all that has to be calculated at why, um equals zero and at y equals toe over X squared. So we calculate this victor at why equals two over x squared and then subtract the same thing calculated at y equals zero. Um, X squared. Um, integrated Respect. Why is just X squared? Y um So we have to take this expression at y equals zero and subtracted from the same expression taking at why two over X squared. Ah, let's simplify the top. So this expression at why equals two over X squared is X cube times two over X squared. This expression at why equals zero is just zero. Right? Then this expression at why equals two over X squared is one half x squared. So I substitute over X squared in place off. Why? So I get over X squared squared, which is equal to four over X to the part off four. Now, this expression at why zero is just zero. I'm not going to bribe the zeros because they don't make any difference. Let's do the bottom. Okay, so X squared times why? But why is to work squared? So X squared councils? Oops. Great. This is not the victor. This is just a number. Then this expression taken at y equals zero is just zero. So I'm not writing it. Okay, so, um X squared X cubed over X squared is just x X squared over X to the power of four. He's just actually. Okay, so we get X times two so two X on top and to over X squared is the second component. All the victor on the bottom is just just to integrated with respect to X um two x integrated with respect to X, it is just X squared two over X squared, integrated with respect to X is just negative to over X. And this victor has to be taken and zero Oh, sorry. At one and at two. So we take this expression at one and subtracted from the same expression taken at X equals two here. This integral to DX is just two x. Okay. X squared at two is four x squared at one is one negative to over X at two is just negative. 2/2. So it's negative one. This at one, is just negative to over one so minus minus two, which makes it plus two. Here to accept to is just four and two x at one is just too. So we get three one divided by two Dividing this victor by two yields another Vector three calls one house, which is the radio's victor off the center of mass. So this completes the problem

Oh, you're doing well. So gutter equations here that formed the boundary of our region. And we're told that a regions down to buy the X axis, which is where we get the wise equal to zero our foundry, this forest X values are up. Lower limit is one. And that problem is to the excise. So make a quick sketch of this. You know that wise included 24 x squares. Gonna look something like that. And then we've got just the X axis strike here. So between one in two with a beer regions, this is our region where thin plate is located. That's what we're looking at. Sofrito find epidemics as our upper boundary and G of X is your lower boundary. This right here the screen lights at the VAX just acquitted two over X square. That's a purse function. The slower bounding hero BG attacks to purchase equals zero. So use assist later on when calculating our mass and center. That's so first we're gonna start by calculating our maps or mass is going to be equal to the integral from a the of our density function times, uh, at the backs. My next year that's de ice. Forget to write this dunder density function were given that that's equal to X square. All right, so now we can So this we know where boundaries Or at least know that our boundaries from here ago from once it's you So is that this is going to be cool. Do 16 year times are delta acts or density functions X squared times ffx remember from before is to her X squared exploded and then my issue of exorcists that's just zero. So we don't have to worry about that when STS these X words cancel out. So you end up with in a role from 1 to 2 just too? Yes, taking this integral. This becomes two acts evaluated at 12 Sequels. First playing in our upper boundary two. This is two times two minus them. Playing in our lawyer Battery one. This is two times one. It's just a four. Discuss to support tends to I'm sorry. Four minus two is equal to two. Sorry. Mass is equal to two. All right, so we can use this to get our center of mass. So first women start out by getting your expel. You for a center of mass just equal to one over our mass times. He girl from a to B over density, function times X times f of X minus ji attacks Yes, all right. And I know that our mass before is equal. It's use the whenever a massive going 1/2 times the interval from one since you were density Funches X squared times acts times we know from before that ffx minus year backs just two over x way the DS Please cancel out like before. So we left of 1/2 10 0 gold promoted to i two x dx this year, since it's a constant can go up front that heels is out with one have to do is just once This is equal toe just the interval from 1 to 2 of axity acts. Evaluating this, you have 1/2 X squared, valued from zero I'm served from once. It's too is equal to first playing in our upper boundary. That's 1/2 2 squared minus plaguing our lower boundary. When have tons of ling squared? She scored us four divided by 22 years. It's going to my X one squared is one divided by tears. One, this is equal to three house that are X value for center of mass is equal to three hounds. All right, so now we can move on to our why Value for center of mass, which is equal to one over our mass times the integral from a to B bird density. Sorry. Their density function over two times have squared of acts minus juice, squared bags, DX. All right, so we can take out this one happened. Forget outfront sits 1/2 times. Our mass is from before it's too. Since one how eyes our masseuse one over a massive remember 20 wanted to of our delta X functions. X squared times at the vex is two over x squared. It's that function square minus year. Backs were just your i zero swear T x sunsari this back down. All right, so this is going to be equal to 14 I'm seeing a girl from Morn. It's you Back square times. Do you scored us War over. Exploited Squared this with the excellent board. Yes. Um, this four. We can bring that out front that multiplies with one port just to give us once that's one or two X squared over extra fourth yaks. This is equal to integral from once it's you extra minus two DX valuing this inability of X minus one over minus one. Evaluated once it's you equals minus one of her acts evaluated from one. Since you seems that being holds you first evaluating plaguing in our upper boundary to minus 1/2 minus. Looking in the low ground one this is gonna be minus one becomes a pluses equals one minus 1/2. Secret of 1/2 these things that are why value for center of mass is equal to 1/2. Now, remember, we know from before that air X value for center of mass is equal to three habits in this story, Here's our finance of these air X and Y values for a center of massive distant, please. All right, well, thanks. And I hope that helps

In this problem Where us to show that total force Acting on an object that is immersed in a fluid with destitute row eyes ro in Tegel A to B road times g times x, w x d x ray is the distance between the surface. Off this fluid to the upper surface of the object on B is the distance between the surface So pitiful er to the bottom surface off this object. All right, we know that force as you go to pressure times area, and that is row 10 g times that times D area. Now let's issue not be have an origin located right here. So 00 Ellis, assume that this is the Y axis and this is the X axis. Now, what we see from this object is that as that that changes so as exchanges. The what changes. It means that if you were to take an area offering to infanticidal strip and if he is to not areas so make X or deadly x times the height G X. So the area What that would change. Okay, Now we know that this is the origin. So this x zero level this is X equals a level and this is X tickles beat level. So the area off distance trip would be w X T X and total area off this whole object than integral w x D eggs. So what is the That's so Roizen material? Poor pretty G is universal. Be found an expression for areas, a function of X here. Now what is that? Well, since we assume this point to be the origin and since we know that that this measured from the reference point and since this is the direction of supposed to backs, we can just used exported that sort of total force would then be in charcoal from, um, well, let's talk about the limits later. Be have density times g times depth. We noticed his ex enemy noted area is WX DX. Let's look about limits as you can see total side of his object This B minus I meaning that, um, the wet will change when X is equal to a and it will be different when X is equal to be, which means that limits after integral will be from a to B. And if you compare this well, what is given, as you can see, we just drive it


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