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Problem 32 Hard Difficulty

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

$ y = x^3 $ , $ x + y = 2 $ , $ y = 0 $

Answer

$(\overline{x}, \overline{y})=\left(\frac{52}{45}, \frac{20}{63}\right)$

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Video Transcript

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'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't need to then perform two separate integrations. So it means that we're gonna write everything in terms off. Why? So we're gonna convert that into X accuse turned route. Oh, why sexy cools down. Why 1/3 and we're gonna let this one s exegesis. Two months. Why? And we're gonna send it, then our area will be in trouble. We'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'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'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'll subtract lower function from the upper functions. So we have two minus What? Let'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're done. What area? Car collision. Now we're into a cockroach export. And why bar? Since we'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'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'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