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An artist has decided to finish their piece of artwork by balancing it on fulcrum and putting it on display: The artwork has constant density and must be balanced a...

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An artist has decided to finish their piece of artwork by balancing it on fulcrum and putting it on display: The artwork has constant density and must be balanced at 1ts centroid. The shape of the artwork was created on computer program then casted and fabricated The following equation was put into the computer to generate the shapey = 4sin(Tz) + bounded by 1= 0.x=2 andy = 0Draw the Lamina in an x-Y plane and put dot Where the centroid should be Show all work and formulas You are usingThe centro

An artist has decided to finish their piece of artwork by balancing it on fulcrum and putting it on display: The artwork has constant density and must be balanced at 1ts centroid. The shape of the artwork was created on computer program then casted and fabricated The following equation was put into the computer to generate the shape y = 4sin(Tz) + bounded by 1= 0.x=2 andy = 0 Draw the Lamina in an x-Y plane and put dot Where the centroid should be Show all work and formulas You are using The centroid is at (€,9). Where Preview y = 3.3 Preview



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Sketch the region between $y=x+4$ and $y=2-x$ for $0 \leq x \leq$ 2. Using symmetry, explain why the centroid of the region lies on the line $y=3 .$ Verify this by computing the moments and the centroid.

Okay. Given wise people to express one. And why is he good? Thio ex punish one squared and were asked by the center for meth. Yeah, it's drawing us out. Yeah. Why's it could explode? One we have. Ah, why? Except that one. Our exit intercepted that negative one. But yeah, you won. You have something like this straight line, and then we have a wise expert. Is one where Hmm? Let's see. Well, we know are little. One point is zero comma one. But its problem here at why is equal to one x minus y squared. It's this This is one. Is you experienced one. Our exit Still, Joe comma one here. So the interject here and we need to find the other point where in which the Intersect. So let's just make them equal to you. You have experts one, as you could to x minus one. It's wade. Well, X minus one squared. Is this X squared? Minus two X plus one. Okay, let's move this to this side over here. So we gotta x squared minus three x. We don't x ex ministry, because it's also the Intersect that extra tickets to go on except It was great and acceptable to Dre. We have Wise Eagle before, so I want to report 12123 Oh, it's a job like this. You were asked on century of this portion here, that's from 0 to 4. So three our exit with three. Our wise for Okay, well, let's start with that. So, um, why is equal to P times in a row? From 0 to 3 of x times? Our top graph, which is our line X, was one minus our bottom graft, which is X minus one squared, which is also equipped its two X squared minus two X plus two X minus one. Yeah, the next simplified It's a little bit So we get negative X squared or one's cancelled and we're left with blustery X the X We can simplify this murder by if you bring in the ex till we get peon A girl from Joe Andre. Uh, negative excuse, Blustery X squared. The ex students is equal to make a fixed to a board over a floor. This three year Tonto, Thank you. Evaluated at three in jail. So we get creative power for gives me 81 over four class three to part three. She's 27 27 times or is one of eight. So he anyone +108 over four. What is that? That gives me 27 over four times. Right now Looking for M X. We're gonna use equation DRI Ethical thio 1/2 p from 0 to 3 Again of my top function squared I think this was one squared minus my bottom function square That's X minus one to cartoons. And two, it's for the ex. All right, expanding this And since I know X minus wanted Part two is this You have x squared when it's two experts one. I'm experimenting to Extras One Okay and experts with you. So that is re X squared Plus two experts one. This is extra power for Linus to execute, plus X squared when it's two x cube Linus R plus for X squared. When it's two x plus x squared wants to expose went simplifying this a bit extra. But before powers of three is negative for X squared gives hard too. 1231236 x squared There's one you get bigger four x plus one. Okay, so rewriting M of X as he could to 1/2 p within a girl. Uh, this. What is this? This entire thing? Wyness This Alexis Jack This minus plus minus, plus minus plus minus. And in combining these two are X squared. Who's in seven? X squared. We had seven x squared, and it's become sport plus two six x question. Okay, so we have you could have extra pair of four was for X cube. The seven ex squid six extra assistant. Is this correct? Minus one. Ah, whips. That should cancel actually. So there's no plus two here. This plus one minus one. And this is minus six plus X squared, which is bigger. Five x. Yes. Oh, this fun. Go back to this next year. Five X squared. And this is positive for exposed to X, which is six X and we have plus one minus one, which is this drill. So this is that DX from me. Double check here. This was a minus. That should be minus my back. Square Tips should be minus and plus 4 to 6. Okay, Yes. So I think the integral this week a negative excuse. About 5 45 What? Thanks for minus I have excuse over three plus X squared plus three x squared Valerie to that. Well, firstly, times 1/2 p and evaluated at Joe's. Okay, let's begin. So get 1/2 p in violation at three. Is there a two? Part five gives me two for 2 3 to 4 to three over five plus three through five history. 81 on the streets of our three times five is a 135 over three. That's 45 plus nine times three. Excuse me, 27. Okay, mine is a revelation. And so did you have X x x x years and it's zero. All right, so let's combine this. We have been able to for three over 5 81 when it's 45 37. Gives me 14 to over 5 14 times five somebody to replace. Hey, but that's also multiplied by 1/2 p. The 72. What about two is 36 p over five. All right now what we found and next it's fine. So that's this. Peeing in a row from 0 to 3 of the top graph, which is X minus one x plus one minus I bottom graph, which is X squared minus or close to X. Like this one, the Ex East to terms cancelled. Here we can combine our action to X. So we get in a girl from zero to drink he of negative X squared close three x x. Okay, that's the question that it affects. Cubed over three plus three squared over two evaluated at three and zero times. Okay, so we get P dreams that are three. 27 over three is nine because in England, nine plus created her two I'm staying is 27 over too. An evaluation of those. So 27 over two months. Nine iss for 1/2 its nine of Jim So nine p over too. Okay, so we're not m is equal to nine peeing over too. And why is it so? 27 pee over four and an X is equal to number. I so our center of mass because you close to ex CNN. Why I see him If you go to m y May 7 p over four. What if I Mm. Which is times to pee or two over 90 canceled too. And an ex just 36 p over five times to over 90. All right, let's simple finest a bit. So we look forward to and this is dying times three. So our knight can't So So we're left with nine or three over two. Archies Awful. Cancel best. This is nine times four. So we cancel off her nines. That's four times to which is it's over. And herpes also cancer this low center of breath.

We're as to sketch the region enclosed by the functions y equals zero y equals X plus one cubed and y equals one minus x cubed and then define the Centro right of the region. So to sketch the region first, let's just sketch y equals zero. Now we're going toe only draw the 1st and 2nd quadrant here. This has to be above y equals zero y equals zero is simply the X axis. Now you want to also draw the function y equals X plus one Cute. This is going to have a triple route at X equals negative one and we see that it's going to have a Y intercept and why equals one? So we're going to have point here it one and then also a negative one. And because this is a cube function, we have that say negative one half. We plug that in. We get why is equal to one half cube, which is 1/8. It's a really quite small. And so you get the idea that this function curves up like this and likewise we have that y equals one minus X, cubed as a triple routes at X equals one has a Y intercept of one. And we have that when X equals one half y is equal to 1/8 So likewise a symmetric sloping line. And this is this green area. Here is our region. We want to find the Centro it off. So to do this, we want to find the moments of the region. First we have that The X moment of this region by Equation three is one half times the density which for a century oId is one times the integral. And this is going to be from negative 1 to 1 of well, not exactly negative one the one we actually have two different into girls here. So using the activity of integral So we have one half times integral from negative 1 to 0 of and then we have the function squared. This function was y equals one plus x cubed. This is going to be one plus X to the sixth DX. Into that, we're going to add the integral from 0 to 1 of other functions squared, which was one minus X cubed square which becomes one minus X to the sixth DX and to evaluate this integral what you want to Dio is use the binomial theorem and multiply these out. And you eventually obtained that this is going to be 1/7. This is a rather lengthy calculation. So I'll leave this to you. We get 173 decks moment and then we have that The Y moment we can see immediately is zero. Now see why Notice that this region is symmetric about the line X equals zero about the y axis. This tells us that the Centro oId lies on the Y axis and therefore that the X coordinate for the Centro oId is going to be zero. This implies, since the X coordinate is, do you? Why moment, Over the total mass. This implies that the Y moment must be zero. So we have that. Why Moment zero Since region is symmetric about the y axis. So with their moments calculated Last thing we need Yes, total mass. So we have the total mass is going to be the density. Mm hmm. Which is one times the area which is the integral Mm. Sorry about that. We have that the total mass is going to be area which is integral from negative 1 to 0 of X plus one Cube DX plus the area of the right half, which is integral from 0 to 1 of one minus X cute t x. And once again, this is a calculation that's a bit lengthy. All you really have to do is we'll play out that China mules and then find anti derivatives. But you see that in the end, the calculation gives This is one half, and therefore it follows the Seine. Troy has coordinates. We have the why moment, over the total mass, followed by the ex moment over the total mass. They calculated that the wine moment was zero, so it follows that this coordinate zero. The other coordinate calculated that the X moment was 1/7. So this is 1/7 over one half, which is 2/7. And so it follows. That's Sen. Troy has coordinates zero

In this problem, whereas, to find a center of mass alteration bounded bikers wise ical each the x y z ical zero X equals zero and X equals one. So that would be the shaded vision. And we usually we know that or you can see the central myself. The subject will be somewhere around here. Now, let's calculated analytically, You know that we first need the area off this, um, region. That'll be area is internal from a to B, then is the limits ffx d x? Not actually the summation off your fifth decimal areas. 10 strips meaning that were forming this infinitesimal blocks with what d x. And we're sending those up where f effects in this case is e to the X and the limits of our internal is that changes between zero and one, so they will be zero b will be that one, and effects is each of the X DX that is each of the X X goes from 0 to 1, sir. The area is then e minus one. All right, now let's cockle and export. You know the export is equal to one over area integral from a to B. Thanks aftereffects Eggs where we noted areas e minus one. So we have won over B minus one inta go from 0 to 1 x times each of the ex d x here. In order to Seoul this integral we're going to use intervention by parts. So we're gonna say that executions U then e to the X d X is equal to Devi. If the X is equal to you than the actual vehicle to d you And if you do, the X T X is equal to DVD. It means that each of the axis it would be so we can end right this export as one over e minus one. We have ex each of the X minus integral from 0 to 1 each of the ex D eggs that is equal to one over e minus one x e to the X minus each of the X, or exchanges between zero at one. And if he plugs you in one in in conflict, guess we find out that export is Look at it at one over. Eat mice one. Now let's calculate Webber, why? Bar is one of our area internal from A to B one. How off to function described affects the eggs that is one over the minus one he have internal from 0 to 1. What health off each of the ex crisis on our be heated to x d x. All right, so what? House is just a constant so we can take this one outside the interval. So you have one over two times a minus one times integral. Each of the two x t x is or entire day with before that would be easy to the two x delight to that. Exchanges between zero and one that we have one or four times e minus one were forthis two times too. So he had one over four times a minus one eastward months one. And we can write each card minus one as what are four times d minus one e minus one, most by a plus one e minus ones will cancel out So that y bar will that be equal to one over or e plus one over four sort of central mass off the system will be down at one or e minus one and plus one over four

Hey, miss problem. We're giving any question of any 1/4 for circle. Why? Didn't give any question is why's it called scourge sexy sex card. If he noted exchanges between 04 and thes e questions are also known and given in a problem statement, we're trying to come play export and white bar for a 1/4 off a circle. All right, since we're going to use be given equations, we first need to calculate D s. And this is our be calculated in order Chocolate D s. We would need to first find the thereto. Ought to give any question which respect him. Thanks. So let's tag narrative off. Why would suspect Activity X would be won over Skirt one over to skirted of 16 lines, expiry months. Bye bye. There was intervention. It's native to X Tuesday against love. So we have that one has negative X delight. I'm skirted of 16 months excrement and skirt of that S O D I D X skirt would done be equal to X squared. Divided by 16 months. Ex skirt. All right, we have everything we need. Less display goes in. Export is equal to one over l what is L l in this case is the circumference. And for a circle circumference is two point times radius. And as you can see, Radius s four side will be to buy times for I've ever given 1/4 of the circles of in to divide this spot for So does your conference off that peace will only be two ply or s o for expert. We have won over length one or two by integral forms here before x times d s where d s is scribbled off one plus thes y d X skirt that is equal to x Great delight by 16 lines Expert The X that is equal to one over to pie trickle from 0 to 4 X times skirt of 16 divided by a skirt of 16 months Ex crime DX How did I get this Second port? Well, all I did was to multiplied this, uh, one by 16 minus explorer. And do I buy the same number? Skirted off 16 is equal to four. And we can just take this one outside the in. Jekyll and right export as, um four over to pie. Internal dessert for ex skirted off 16 months Ex card here, We're going to say that 16 months Next. Carter's our new variable. You saw that native to X E X would be to you and X t x Could dummy continue native d'you or two? And since dishonorable one X is zero you 16 and one access for you. The zeroes are the limits of formula In title will be then from 16 too zero he have negative d'you deal. I skirted a few soling Thus we find expert to be equal to eight or pie LSD chocolate white bar Why bar is worn over el again one over to buy integrate forms here to walk For this time we're gonna let's buy Why buy deep s we know what dances and we know why as a function of X status given to us So that'll be 16 months Ex crane multiplied by, uh skirted up d s, which is this part in which we calculated it to be as this term. So that would be skirted of 16. Delighted by skirted off 16 minus expert the ex Nazis will cancel out skirted off 16. It's four. So we have four over to pie off ex were exchanged between 04 That is an equal to 16 over to pie, which is eight or apply. So from this, we can export to be equal to eight over five. Why part to be equal to eight over pi.


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