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Find the centroid of the region bounded by the given curves.

$ y = 2 - x^2 $ , $ y = x $

$(\overline{x}, \overline{y})=\left(-\frac{1}{2}, \frac{2}{5}\right)$

Applications of Integration

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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's find the intersections that first. So we'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'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'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'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'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'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's two months ex scraped. If you must buy that by X, we would get X times. Well, let'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'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'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'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'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.