solve-the-given-problems-by-integration-find-the-x-coordinate-of-the-centroid-of-a-flat-plate-that-c

Question

Solve the given problems by integration.
Find the $x$ -coordinate of the centroid of a flat plate that covers the region bounded by $y\left(x^{2}-1\right)=1, y=0, x=2,$ and $x=4$

Doruk Isik · Accepted Answer

in this problem, whereas to find central region bounded by occurs wise vehicle to two minus x cried and wise ical x It is a region, the shaded region. We can see that those cars intersect at to Florence. Let&#x27;s find the intersections that first. So we&#x27;re gonna set to minus X cried to be equal to x. From this we see that X cried plus X ministries zero That is to one expects plus minus Sorry to me that those intersect when X is equal to nated to annex to eagle one. So we have to intersection points or two intersection levels. That is, one X is equal to negative too. And when X is equal to one now, let&#x27;s find the area off that region area would be integral. Um, something times DX. Why wise dx? Well, the idea is that we are forming 10 strips with with dee eggs and were something those up and that would give us the whole area. Since we&#x27;re something those off, we need an integral and we see that exchanges between negative two and one sort of limits will be negative one for that region. As you can see, we have two terms and the upper chur. Um Ms case is two months. Sex, crime and lower turn. That&#x27;s what Ex studying Targo would then be to minus X cried minds x d X, and tied everything off. That would be two x minus X cubed over three, minus X skirt over two and exchanges between Negative, too, and want and calculating best. We see that the area is 26 27 over six. All right now, let&#x27;s talk about export expression for expires one over area integral from A to B x Times F effects or the upper function minus still over. Function the ex. You know everything that we need. So let&#x27;s play. Everything in the area is 27 over 61 over. That would be six over 27 inta go from negative 2 to 1 because that is the limits off X X times if effects Where the upper function again. It&#x27;s two months ex scraped. If you must buy that by X, we would get X times. Well, let&#x27;s write it anyways. Exc rage minus. We have X D. X that has six over 27 integral from negative to do one. We had two eggs, minus X cube months. Ex crate DX explore would then be six over 27 X grade minus X in fort over four minus X cubed over three where exchanged between edited to and what if you plug in and two and one. If you do the calculation, be fine. Export to be negative on health. Now let&#x27;s call it please. Why Bar expression for Bob Barr is one or area inter. Go from a to B one. Health off upper function straight minus lower function Squared The Ex. Let&#x27;s plug everything that we have in y bar would be six or 27. That is what our area integral from negative 2 to 1. What health off or per function square. So that is to minus X grade. Sprayed minus X squared DX. Yes, well, cats loud. You&#x27;ll have 10 3 or which 17 in jungle from negative 2 to 1 four minus five X squared plus extra fourth e x. He had four here. All right now why bar would then be equal to three over 27 four x, but it&#x27;s five x cube over three plus extra 50 over five for exchanges between negative two and one. Let me do the calculation. We see that why bars equal to three over five sort of central mess off the total whole region would be that at negative one health and two over five.

Doruk Isik · Answer

in this problem where has to find a central region? Bounded Bikers excuse Why Skirt and line X plus wise too. We first need to calculate the area off the region. And when calculating the area, we have two options. Big leader ofour blocks, 10 strips, but with the eggs so in extraction. But the problem? What? That would be, um, while interrogating. So while something those two strips out you would first in tracked What disc? Er right here. As you can see, that is why X equals y scrape. And that after a points after this intersection point he would be interacting with this line, which is exploits Y is equal to two. So we would need to do we would need to have two separate integrations and inertia Ever work that Mr Forming this vertical strips were gonna form horizontal strips like this. What? That we will be interacting with X equals y skirts or disk herb and, um, this line simultaneously. Now this 10 strips they have the wit off the Y sort of area would be this length times with or this length times the height off the Juan that since we&#x27;re something goes up we need to integrate it. And in order to cope pity or alert integration, we need to find the limits off that integration. As we can see, the Oscars intersect at two levels. This is the first level, and this is the second interrogation port. And that will be the bounce or D limits off our integration. So if you first need to find those points in order to find those less, find the intersection points to do that, we&#x27;re gonna set. Um, first of all, let&#x27;s write destroyer in terms of Why so I said at exit A goose to minus y So we&#x27;re gonna set wife&#x27;s courage to be equal to mine. Swine. From this we see that then once grade plus why minus do should be zero. So to one. What? Why? That could be plus minus so days. Those intersect when why&#x27;s it goes negative water and wise. It was to start going into intersection levels are Why&#x27;s it going negative one And what was the cooze? Why is equal to one Saudi? Interrogation limits will then be negative one. They head into one and since we&#x27;re interrogating, would suspect a Y in why direction are up for sure. Well, then be to minus wine. So this would be the upper function or f f y and ex schools. Why would be lower function? Or that would be joke y. So for the area you&#x27;ll have fo Fireman&#x27;s Joe find out is to minus y likes y scraped d y. That is negative. 2 to 1. Uh, well, every king get rid of the integral and just find. Been trying every soft dock that would be too y minus y squared over. Two. Let&#x27;s white cube over three or one chance to between two and one and plugging negative two and one and and inflating. Be flying the area to be nine over, too. All right, now, let&#x27;s got quiet. Explore, since you are integration is now with respect to why D formulations for X Bar and y bar are flip so inert to Cochran expert. We&#x27;re gonna use the relation for why bar what I&#x27;m saying? This that will be won over a inter go from a to B 1/2 off. That&#x27;s great. Why, but it&#x27;s g skirt of why d y you have everything that we need. So let&#x27;s display that in expert would then be won over area where areas nine over to sew one over. That would be to over nine. And we know that limits our betweennegative to it. One saying juggle from negative turn one one house off effort. Uh, X squared. Why, which is we know that effort flies to minus five sleepy square that we get four minus four y plus y squared minus discredit for where? We know that g o wise one square. So if you scared that people got park to the fourth, do you wanna know what health is? Just a constant? So these will cancel out and entitled editor for this argument inside the intractable will then be one overnight we have four y minus two y squared plus one cube or three minus white in effect over five or white changes between negative two and one and cackling not be fine export to be equal to 1.6 l s cock late. Why bar again? Is that sadly formulations for explain award by our flip So we conquered wipe are using whatever area integral from a to B. Why times f f y minus you. If I do, I that is one of our area is to over nine interrogating from negative to one. Why times to minus y minus y sck rage D y that is to over nine integral from negative to 12 y minus y scribe mites, y que deal I an entire day with afford for that would be to over nine. Weiss cried minus y cube over three months. Wanted a fort or four? Or why changes between negative to anyone? And if you pluck those numbers in a duty of elation, we see that white bar is equal to negative 40.5, so the center of mass off this region would then be at 1.6 and negative 0.5.

Doruk Isik · Answer

in this problem. Where is to find the century outrage in form writers? Why&#x27;s it called Ex Crane and Ex Eagle? Why screamed? So let&#x27;s first of all right, this curve as Why&#x27;s he going to skirt X? All right, and let&#x27;s first find the intersection of those two cars, meaning that we&#x27;re going to say that X Cried is equal to school of X. From this, they see that desk holds when x zero and when x is one. So it means that thes two curves intersect at Execute Zero and Texaco&#x27;s one. All right, so now let&#x27;s first find the area off this shaded red region. The idea is that we&#x27;re taking 10 strips with with off DX and now we&#x27;re something goes up to calculate this whole area. And since we&#x27;re serving those up, it means that we&#x27;re gonna need intra go as you can see exchanges from 0 to 1, so the limits will be from 0 to 1. Also, as you can see here, skirted of X as dicker that is on top. And this is the wanted bottom X cracked. So we&#x27;re going to set up the area will be done. A picture minus bottom. Sure. The X that is your three extra three over two minus X cubed over three, where exchanges between zero and one. And when we talk like this, we see that the area is one or were three. All right, Now we&#x27;re ready to Kat Kat. Export export is equal to one over a integral from A to B X. Times effects minus g affects the X. We have everything that we need. Area is one over three. So one over one under three would be three ingi go from 0 to 1 X times skirted off eggs and don&#x27;t forgot. Skirted of X is equal to X to the one health. So if you multiply that by X, we would get extra to three over two minus. Now we&#x27;re multiplying X by g x where GI effects is expressively here X cube D X entire ever talk that would be three times to over five extra five forward to minus Exeter forth over four where exchanged between zero and one. And when we plugged want and Syrian and cock Wait, that&#x27;s we find out the export as at 9 20 now less cockpit. Why bar you know that y bar is equal to one over eight. Internal from A to B what have off f squared X minus g scarred of X d x. That&#x27;ll be then one or area is very integral from your one if affects the skirt effects of skirt off. Done would be ex GI effects expert. And if you screamed at me, we got extra t forth. D x, this is four. All right. Now we have three. An entire day with folk. That would be ex skirt over too. But it&#x27;s extra to fit or five were exchanged between zero and one. We don&#x27;t get wine. Bar s three times one over two, minus one over five. That is equal to nine over 20 sort of center off mess off this total system. Would region would be that aunt nine over 20 and nine over 20

Doruk Isik · Answer

in this problem. We&#x27;re trying to find Central plan. Soft region bounded by wise. You could skirt if x y zero and Mexico&#x27;s four. So that is the shaded region. Officially, um, the center of mass should be somewhere around here. That is our estimate, Nellis. When the exact quarter, it&#x27;s an ethically be no, not for you. First need to calculate the area that region an area we&#x27;ll be in triangle some areas off that 10 strips with with dee eggs. So we&#x27;re going to some of those up, meaning that wagon is integral. We&#x27;re going to integrate this function f effects D x from X equals zero to execute us forces your four FX DX where if effects is credited, effects will be how integral a 0 to 4 skirted effects the ex Remember, skirted effects is extra t um one health. So it means that this integral will then be become to over three extra three over to where it changes between zero and four. And from this we see that then the area is it. Go to 16 or three. Now let&#x27;s go clad. Ex bar export is one over area integral from A to B. where. A zero bees for X ffx DX No, let&#x27;s play everything in the known at Area 16 or official one or area would be three old 16 inta go from 0 to 4 x skirted off x d x Dennis three over 16 Inter go from 0 to 4, thanks to the three over to D eggs the entire Garrett off. That would be X to over five extra to five. Forward to where exchanges between zero and four so we would find expire as three over 16 times to over five months. Fly by 32 which is four to the five. Forward to that is to sow The X coordinate would be at 12 or five in order to find a Y prime. We have are in Turkle as one over area again, which is three over 16. We know it from export parte integral from 0 to 4. Be designed How, What? Health Off the function squared where if effects as started effects. So skirted affects scribe D X that has three over 16 times one. Health is just constant, so we can take this one outside the integral integral from 0 to 4 x be eggs that is equal to 34 32 x skirt over to where I&#x27;d change it. T 04 From this, we see that we had three over 32 times 16 over too two, so there&#x27;s three or four. So the central mess off the system is act on 12 45 and three or four.

Doruk Isik · Answer

in this problem. Worse to find central region formed by curious why she called X Cube. Explosive wise. You could do. And the X axis that would be de shaded region. No, we know that in order to find central reefers need to calculate the area. And the idea of calculating area is to form 10 strips. Uh, what d x and some of those up until we get the whole area. Now, if you were to form 10 strips in extraction, as you can see, leave it first interact with why&#x27;s ical X cube on this part? On the second part, we would interact. What? Why did you go to money? Sex? And this would be the second part. However, if you were to four chairs, but what off zoomed and what went off d y then we would be interacting with wise ical x cubed. And why is it going to minus sechs simultaneously so you wouldn&#x27;t need to then perform two separate integrations. So it means that we&#x27;re gonna write everything in terms off. Why? So we&#x27;re gonna convert that into X accuse turned route. Oh, why sexy cools down. Why 1/3 and we&#x27;re gonna let this one s exegesis. Two months. Why? And we&#x27;re gonna send it, then our area will be in trouble. We&#x27;re something those 10 strips up. However, if you first need to determine on the limits off this integral, we know that Weikel zero is well limits. So that is why I called zero level. Now we need to determine this exact point. How do we do that? If he said why cause or why I didn&#x27;t want her to be equal to two Mighty Swire. This gives us why coz one as a solution. And this implies that this level, it&#x27;s like a small level. So inter goal will be between zero and one now, since we switched to integration in what direction are up richer are a perfection will be that dysfunction. So this is f f y, and dysfunction is still flying. So what chocolate and area will must fly the upper function or we&#x27;ll subtract lower function from the upper functions. So we have two minus What? Let&#x27;s watch the 1/3. The wire area will be to y minus y skirt over too minus three over four. Why did the former the World Watch Institute 01 from this We found the area to be three or four. Now we&#x27;re done. What area? Car collision. Now we&#x27;re into a cockroach export. And why bar? Since we&#x27;re integrating with respect, why explain Why buy flip now? So the relation for white bar would be T uh, whatever. In jungle from won over area internal from eight to be wine times f y minus geo. Why do you are that is equal to one or area. Where areas three or two. So whatever, that would be four or three in Tegel. The limits of why is wise your toe? What wants with 02 one? We have to wine minus y squared minus y 2 to 4 with three d y. So why bar would be equal to forward? Three. Why? Scribe? Minus y cube forward three minus three over seven. Watching the seven over three. Where white things between zero and one. And if you plug one in? Uh, since we know that one was your this whole thing is zero. We find y part to be equal to 20 over 60 three now less cockroach export export would be cooked one over area inter go from a to B 1/2 off F squared off. Why minus cheese. Carter. Why the wine that is four over three Inter girlfriend 01 to minus wine. Squared minus. Want three or two D one that is equal to well said at one Health. Let&#x27;s not forget it. Diesel cast laugh. So we have two over three. Integral from 0 to 14 minus four y plus y squared minus. Why to the four over three d lie. It&#x27;s actually two or three. All right, so the entire derivative for that would be two or three four y minus two y scribe plus wine cube over three, minus three over five Wine. Five. Forward three hour. Why changes between zero and one? And if you do the calculation refund expert to be equal to 52 over 45 so we see that then center of mass off the system will be then at 52 or 45 and 20 over 63

Problem

Solve the given problems by integration. Find an …

Problem 32 Easy Difficulty

Solve the given problems by integration.
Find the $x$ -coordinate of the centroid of a flat plate that covers the region bounded by $y\left(x^{2}-1\right)=1, y=0, x=2,$ and $x=4$

Answer

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Anna Marie V.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus