Problem 29 Hard Difficulty

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

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

Answer

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Video Transcript

in this problem. Where is to find the century outrage in form writers? Why's it called Ex Crane and Ex Eagle? Why screamed? So let's first of all right, this curve as Why's he going to skirt X? All right, and let's first find the intersection of those two cars, meaning that we're going to say that X Cried is equal to school of X. From this, they see that desk holds when x zero and when x is one. So it means that thes two curves intersect at Execute Zero and Texaco's one. All right, so now let's first find the area off this shaded red region. The idea is that we're taking 10 strips with with off DX and now we're something goes up to calculate this whole area. And since we're serving those up, it means that we're gonna need intra go as you can see exchanges from 0 to 1, so the limits will be from 0 to 1. Also, as you can see here, skirted of X as dicker that is on top. And this is the wanted bottom X cracked. So we're going to set up the area will be done. A picture minus bottom. Sure. The X that is your three extra three over two minus X cubed over three, where exchanges between zero and one. And when we talk like this, we see that the area is one or were three. All right, Now we're ready to Kat Kat. Export export is equal to one over a integral from A to B X. Times effects minus g affects the X. We have everything that we need. Area is one over three. So one over one under three would be three ingi go from 0 to 1 X times skirted off eggs and don't forgot. Skirted of X is equal to X to the one health. So if you multiply that by X, we would get extra to three over two minus. Now we're multiplying X by g x where GI effects is expressively here X cube D X entire ever talk that would be three times to over five extra five forward to minus Exeter forth over four where exchanged between zero and one. And when we plugged want and Syrian and cock Wait, that's we find out the export as at 9 20 now less cockpit. Why bar you know that y bar is equal to one over eight. Internal from A to B what have off f squared X minus g scarred of X d x. That'll be then one or area is very integral from your one if affects the skirt effects of skirt off. Done would be ex GI effects expert. And if you screamed at me, we got extra t forth. D x, this is four. All right. Now we have three. An entire day with folk. That would be ex skirt over too. But it's extra to fit or five were exchanged between zero and one. We don't get wine. Bar s three times one over two, minus one over five. That is equal to nine over 20 sort of center off mess off this total system. Would region would be that aunt nine over 20 and nine over 20