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Numerade Educator

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Problem 27 Hard Difficulty

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

$ y = e^x $ , $ y = 0 $ , $ x = 0 $ , $ x = 1 $

Answer

(0.58,0.93)

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Video Transcript

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