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Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y=2(x+1), y=3 x+2,$ and $y=8$

Calculus 2 / BC

Chapter 26

Applications of Integration

Section 4

Centroids

Campbell University

Baylor University

University of Michigan - Ann Arbor

Boston College

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okay, were given this graph right here is why X cubed minus six X in this graph right here is equal to ex Murder Street X squared is equal to white. Let's find our intersection points because we're asked defined area of this petition. Be tree. Okay. What's it from equal to each other yet? It's minus three x squared. Is it X cubed? Minus six x. Okay, now moving them all to one side. We're gonna x cubed. Minus four plus three X squared minus six X minus eight people is out. How can we find a route based on this? We already know two of them. One of them think that one and the other one is two. So let's go with exes. X minus two and long Divide and find our other two words. Okay, we get X our X squared over here. So this is execute minus two X squared. Jack, this We get five x squared here when it's X X. Okay? Yeah, doing it. That so? This is no. Five x by the X squared minus 10 x To check this, we get four x minus eight. Annan plus four get four x minus a projecting this. We get so Okay, so our new equation is X minus two times X squared. Housewife, ex plus four. And we can factor that because we know one's negative one. That's experts one. And was this expose for I believe, with strike our expanded, it'll get X squared plus four x x plus for yeah, that Dubai. Okay, so our roots are excessive. Multitude. I'm here in section 4 30 is exiting good to negative one in negative for But it's point over here that we didn't know was negative for all right. Now working with this, let's take the in a row from Negative Forge. Negative one of the top graph, which is excused. Money six X, uh, checked our bottom bath, which is three x R eight minus three. Explode the X plus in a girl from negative. One, 222 overtop craft, which is It's minus three X squared minus our bottom graphic X cubed minus X x. Your ex. Well, yeah, actually, from looking at this now, they gave us the points for roots. I should have known that this was a major for okay. Anyways, simplifying this a bit. What? Yeah, plus the exquisite minus six X when it's eight. The X plus in a girl from Bigger one. You too. Um, I'm gonna execute minus three x squared plus six x Those eight. The external instinct in a growing list. Well, that's just Hey, I'm an extra before or for Phyllis three X Tuned off. Three assists. Executed minutes six x squared over two inches. Three X squared minus h x. You violated that. Make one good for plus negative X for over floor minus X cubed minus four plus three x when it's when it's our cost paychecks. So they're the same thing, which is, uh, you're signed, but different. Evaluated at two and negative. Hey, I'm running over here. Your ex For what? That's execute minus three x squared when it's aged. Excesses, Evaluated ads. You got a four and good one, but the opposite signs evaluated at negative one and two getting up again. They have one yet one over four minus one one is three plus thanks minus evaluation and negative for we get Make it four to Power of four. Does that give me 256 but it by 4 64 Plus I think the four to her straight. Just English 64 minus 34 To show you get 48 minus plus eight times for religious with 32. Okay, now a valuation. Uh, too. You get two to the par four, which is 16 over four. Yeah, the water by four. So I got confused, okay before, But this is a negative science. It's gonna to part three, which gives me negative eight plus 2 +5252 witches before times three, which is 12 us two times a day, but 16 and evaluation. And make it one for this is Linus. You have one over four plus one minus Berta plus three X. So it's three when it speaks. Okay, now, combining these that's seen its combined a constant. So we have negative one minus three and plus eight minus 64 plus 64 plus 48 minus 32. Hottest over here. Minus four minus state plus 12 plus 16 minus one. What is three plus state? We get 40 and then our fractions are 1/4 over 14 which is gives me 1/2. So our area is actually what a 0.5

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