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Numerade Educator

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Problem 50 Hard Difficulty

Let $ \Re $ be the region that lies between the curves
$ y = x^m $ $ y = x^n $ $ 0 \le x \le 1 $
where $ m $ and $ n $ are integers with $ 0 \le n < m $
(a) Sketch the region $ \Re $.
(b) Find the coordinates of the centroid of $ \Re $.
(c) Try to find values of $ m $ and $ n $ such that the centroid lies outside $ \Re $.

Answer

(a) $f(x)=x^{\wedge} 3$
$g(x)=x^{\wedge} 8$
(b) $=\frac{(n+1)(m+1)}{(2 n+1)(2 m+1)}$
(c) $m=100, n=99$

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Video Transcript

in this problem. We're given that X or is between zero and war and were given to functions off. Thanks. Well, one is extra to power em. Another one is extra power. And here we are saying that M is greater than And so the idea is that since X is a number between zero and one, since it is less than one as the power gets larger, devalue have to function well, I should get smaller, so forgiven millions X x to the A. M, which is disc. Your should give smaller value next end. That's why De Graff looks something like this. And this is a region formed. I, uh, are bounded by those who cares. All right. So in part we were trying to find the export And why bar off the system? Be first, know that we need to find the area off that region. We know that exchanges between zero and on so live in juggle from 0 to 1 Ah protector minus overture. But what is the upper care? We just said I since times greater done wanted. Since Sykes is less than one being on it as part against larger devalue will get smaller, so extra the annual than be the opera function. F effects are extra Dechaine on this extra the M D X that would be equal to X and plus one. Do you like n Plus one minus eggs and plus one divided by M plus one where exchanged between zero and one from this. Be fine D area to be and minus and delighted by and plus one times and plus one. All right, let's find X bar. Uh, we know that expert is equal to one over area internal from a to b x Times FX minus jfx he X We know everything that we need. So let's photos and export down will be won over area or area as dysfunction. So whatever that would be in plus one times at M plus one divided and Minds and Inter go from 0 to 1. You know that extra? The end is F effects and excellent. M is GOP XO. If you must buy those to be, have extra n plus one minus extra M plus one e x, and we can let this one as and plus one you hide n plus one or EM minds and and we're gonna multiply that Bye X to the n plus two divided by in plus two minus extra m plus two. Delighted by M plus to our exchanges between your worm. From this, we see that export is then equal to and plus one times M plus one. Do you live by in plus two times 10 plus two. All right, be no have our expert Now we're going to copulate White bar. Why? Barb will be won over area interrogate for May to be one health off f squared X minus G skirt x d x We know that actually end is our facts that I don't be n plus one times and plus one over at minus one. This is one of our area, part in terrible from 0 to 1. 1/2 is just to Constance, where we can take this one outside intricately, right. This one s This will have exited to end minus exit to M e X. It is n plus one tips and plus form. You got two times and minds and wants blood by Exeter to then plus one. Do you like to end? Plus one minus Excellent too. And plus one delight by two M plus one or exchanged between us here on one and from this be fine white bar to be equal to and plus one times and plus one Do you want it by to end? Plus, born tends to enforce. Born. No, we don't want part. Be nowhere. Party for this one. We're us. Approximate eminent. Um, the ideas too set and very close to each other. And we've owned those to be as large as possible. Sorta, this region will get smaller. Okay. Yeah, we know that. Um, so since she does, we want those to be as close as possible. What's that? Tend to be equal and plus one. From this? We see that then export is equal to n plus one divided by M Pless three. And let's take and to be 100 m to be 101 From this, we find X bar to be 0.9805 and doing the same for y bar plugging those values and refund. Why work to be 0.25 24 Now, we need to, um, compared this value with the Functions Valley. So now we can see whether descent toyed is outside or inside that region. White bar We know that what is extra t m for one function wise extra. The end for the other function that is eccentric is that 0.98 is Europe I At this 0.980 fire, we take him to be 100 and enter. Sorry, we take and to be 100 m to be one of one. From this, we see that both wise are about 0.14 Since both wise are less than why far we found which means that this region or D centrally lies outside the region.