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Numerade Educator

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Problem 38 Hard Difficulty

Use a graph to find approximate x-coordinates of the points of intersection of the curves $ y = e^x $ and $ y = 2 - x^2 $. Then find (approximately) the centroid of the region bounded by these curves.

Answer

$(-0.37,1.22)$

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Video Transcript

in this problem Where has to find a central region? Bounded Bikers wise ical each day Externalizing two months expert. That would be the shaded, uh, region Approximately be. Those two cares intersect. One X is equal to make it a 1.30 on. So we have two X levels that is executing the 1.31 in our second intersection or second acceptable as when X is equal to about 0.54. Let's calculate to area officiated region area would be on the ideas to form 10 strips with with DX and some tea areas up until we find the area that'll region. So we're gonna interrogate it. Since we're something goes up, you know that exchanges between negative won't want 30 war and 300.54. And as you can see, this'll is our upper function. FX or FX is then, uh, to minus X cried. And this is our lower function. And this is the geo ex. So, you know, about 40 area. We have intergalactic if X minus two effects, so that would be to minus X cried months. You to the ex D eggs the entire day or two off. This would be two x months excuse or three months Each of the ex were changes between negative won't 30.31 point 54 from this be fine area to be about 1.40 five. Now we know that for the extent with expert will be up for one or area in terrible from A to B or a CZ negative won't put 31 be sport 54. He have X times FX minds gee effects the ex. We hear everything that we need less black. Those in export will be one forward one point for 51 over 1.45 integral from negative 1 31 2.4 Yeah, we have X times affects words your pants where we know what FX is. This is Fairfax and this is your back. So if you multiply those my expertise to get two eggs minus X cube minus x e to the x x This term, it is easy to find anti director. This is easy for this one beginning of suppression off or we're going to use the interrogation. What parts? So we're gonna name X as you so d X will then be able to see you and we're gonna name E x d X to be t fi sort of heat index will be vey by performing integration monk parts. We have an expression for export as one over 1.45 expert minus extra the fort or four minus ex eating X minus each day X Where are things with you? Negative. 1.31 points. Before calculating this, we find export to be about negative 0.37 Ellis chocolate y bar being ordered A white marble booked for water or area Interview for me to be one health off F squared minus G skirt E X. That would be one over 1.45. Integral from negative one for 31 2.54. What? How off FX is two months Xcor. So we have two months exc word squared minus each of the excursions to each to two X the eggs. Why? Bar is an equal to. First of all, we can just take this one house outside intervals. Since that's just a constant we have, um, 0.345 witches. What? Half off? One of wrong 10.45 um, internal from negative. 1.31 2.54 four months. Four x cried plus extra afford minus eating. Two. Ex The Ex An entire narrative for that would be 0.345 for X months four or three X cube plus extra fifth over five, minus Egypt into X over to our actions were too negative. 1.31 point 54. Let me do the calculation. We find Y Bar to be at 1.22 Saudi center, off mass or descent right off. This whole region will be at negative 0.37 and 1.22.