Find the centroid of the region bounded by the given curves.
$ x + y = 2 $ , $ x = y^2 $
Applications of Integration
in this problem where has to find a central region? Bounded Bikers excuse Why Skirt and line X plus wise too. We first need to calculate the area off the region. And when calculating the area, we have two options. Big leader ofour blocks, 10 strips, but with the eggs so in extraction. But the problem? What? That would be, um, while interrogating. So while something those two strips out you would first in tracked What disc? Er right here. As you can see, that is why X equals y scrape. And that after a points after this intersection point he would be interacting with this line, which is exploits Y is equal to two. So we would need to do we would need to have two separate integrations and inertia Ever work that Mr Forming this vertical strips were gonna form horizontal strips like this. What? That we will be interacting with X equals y skirts or disk herb and, um, this line simultaneously. Now this 10 strips they have the wit off the Y sort of area would be this length times with or this length times the height off the Juan that since we're something goes up we need to integrate it. And in order to cope pity or alert integration, we need to find the limits off that integration. As we can see, the Oscars intersect at two levels. This is the first level, and this is the second interrogation port. And that will be the bounce or D limits off our integration. So if you first need to find those points in order to find those less, find the intersection points to do that, we're gonna set. Um, first of all, let's write destroyer in terms of Why so I said at exit A goose to minus y So we're gonna set wife's courage to be equal to mine. Swine. From this we see that then once grade plus why minus do should be zero. So to one. What? Why? That could be plus minus so days. Those intersect when why's it goes negative water and wise. It was to start going into intersection levels are Why's it going negative one And what was the cooze? Why is equal to one Saudi? Interrogation limits will then be negative one. They head into one and since we're interrogating, would suspect a Y in why direction are up for sure. Well, then be to minus wine. So this would be the upper function or f f y and ex schools. Why would be lower function? Or that would be joke y. So for the area you'll have fo Fireman's Joe find out is to minus y likes y scraped d y. That is negative. 2 to 1. Uh, well, every king get rid of the integral and just find. Been trying every soft dock that would be too y minus y squared over. Two. Let's white cube over three or one chance to between two and one and plugging negative two and one and and inflating. Be flying the area to be nine over, too. All right, now, let's got quiet. Explore, since you are integration is now with respect to why D formulations for X Bar and y bar are flip so inert to Cochran expert. We're gonna use the relation for why bar what I'm saying? This that will be won over a inter go from a to B 1/2 off. That's great. Why, but it's g skirt of why d y you have everything that we need. So let's display that in expert would then be won over area where areas nine over to sew one over. That would be to over nine. And we know that limits our betweennegative to it. One saying juggle from negative turn one one house off effort. Uh, X squared. Why, which is we know that effort flies to minus five sleepy square that we get four minus four y plus y squared minus discredit for where? We know that g o wise one square. So if you scared that people got park to the fourth, do you wanna know what health is? Just a constant? So these will cancel out and entitled editor for this argument inside the intractable will then be one overnight we have four y minus two y squared plus one cube or three minus white in effect over five or white changes between negative two and one and cackling not be fine export to be equal to 1.6 l s cock late. Why bar again? Is that sadly formulations for explain award by our flip So we conquered wipe are using whatever area integral from a to B. Why times f f y minus you. If I do, I that is one of our area is to over nine interrogating from negative to one. Why times to minus y minus y sck rage D y that is to over nine integral from negative to 12 y minus y scribe mites, y que deal I an entire day with afford for that would be to over nine. Weiss cried minus y cube over three months. Wanted a fort or four? Or why changes between negative to anyone? And if you pluck those numbers in a duty of elation, we see that white bar is equal to negative 40.5, so the center of mass off this region would then be at 1.6 and negative 0.5.