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Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles (from Exercise 39) and using additivity of moments.

$\left(0, \frac{1}{12}\right)$

Calculus 2 / BC

Chapter 8

Further Applications of Integration

Section 3

Applications to Physics and Engineering

Applications of Integration

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in this problem. Worse to find central Gulf region formed by two triangles and a rectangle. Now let's say that this is our triangle number one. This is trying the number two, and that is directing. Let's find the area's off those region separately. So we have a triangle. One that is two times two delight back too. So that is the sidemen. Split by height, divided. Fight two. Data vehicle to Same for trying the number two you have two times to do. I got to that. Izzy constitute in there. Um, listen, H one this point. And the area of direct angle is the Nico too. Four times swarm. That is equal to for All right. We know that central off a rectangles are guided at his cement access. So this rectangle Semitic with respect its horizontal lines And with respect to this, why access this word, Alexis, which happens to be the y axis. So this would be central office triangle in accordance with them, be zero and, um, negative 0.5 since total hide after red candles. Negative one. All right, now, let's find the central off the triangles. Now we know that Dez line separates or device this link into equal some planes. It means that this is a median. And we know that central will be located. Intersection of mediums and the distance from deaths were techs. To this point will be, uh, two k by K. So the total length will be treat Katie, and we know that is total hide for both triangles is beacon too to that has given. So we know that then three k is equal to two. So from this we see that then Casey, go to two or three. And if case two or three and if discordant off this point is negative zero and it's cordoned off. The sport is 10 in accordance of the central right for triangle one for trying to one recorded off the, uh, central. It will be done, and explication will be attended to one. And why curtain it will be at two over three. So for triangle to do one on the right and x coordinate or lex location will be one. And while the question will be to over three Okay, Now we know that am supp y. So why moment is able to m times X. I now being also known at Mass is equal to or mass is proportional to density. Times area less issue Not both have same destinies, so this implies that mess will be proportional to the area of the region. So using that, we can calculate M setbacks and m sub y. Let's student less time from him. Someone now we're gonna suck form the first triangle The area first trying over he founded is too. Now X coordinate is negative one plus the second triangle X coordinate This one plus direct angle area freak titles for an ex coordinated as equal to zero. From this, we see that then m sub y is equal to zero. That's why I am Cybex so massive First triangle or the area is too. Why coordinate is equal to to over three same 42nd triangle areas to walk Ordinance is at two or three four His area of direct I go and why quarter is that native 25 from this reefer and some eggs as to over they combining these be know that X far is equal to em some why divine by total mess or, in this case, total area that will be zero and what bar is equal to them subjects divided by total mess or a total area, and that will be equal to two over three. Do you mind my total area? His area to trying us indirect angle that is too earthy, divided by eight. So why bar is to over 24? That is well over 12 so decentralized or center of mass of this whole system will be done. It zero mental ward over 12.

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