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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.

$\left(\frac{1}{12}, \frac{5}{6}\right)$

Applications of Integration

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in this trouble, whereas to find a century of this region formed by a wreck tunnel ended triangle be first full. Know that um, mess often object, is proportional to Space City and its area. Let's issue not both regents at the same density, so it means that mess will be proportional to this area. Let's first for calculate the total area that will be area of direct angle plus area for triangle and that will be equal to two times one. This is the area of a rectangle plus area of a triangle. Will be science multiplication, delighted to sort of total area will be equal to then four. All right, now we know that why moment is equal to summation off M times X and X moment is equal to summation off them times y I What we want right now is to individual a determined central messes off those object. We know that for a rectangle central mess sent what will be located at the 70 excess. So this shapes rectangle somatic expected this access in extraction and estimated respect to this work declining. Why directions? So this will be the central and the location will be That is negative. 0.5 that it's one. So the coordinates off the central will be negative. Point five and one. Now, let's find the central right off two triangle. We know that this triangle is a such a median, which devise decided Niko links. Um, so the central we look at it somewhere here and me, not at a median. The proportion is two K and K. So we're gonna set out the total length is three K. Since we know that length off this side is equal to the length of this, this is a symmetry point and accordance off. That will then be one and one. How do I know that? Well, the cordons off this point is zero by two. Coordinates off this point is to buy zero. So I just added X coordinates and divided by the number of points, which is to an added y cordons and divided fighting up off points. All right, now this is the origin 00 we know that from this distance to distance and so in extraction was there were interested in the distance between the origin and this point is one, and that is equal to three k. So it means that Kay some eggs won't say is equal to 1/3 Aminata central. It is located two K away from origin. So the X bar for that triangle will be at 302 L s said that we're interested in why Direction only We know that the difference in walk ordinates is also one. And that is to total as pick up a threesome. A case of wine this time. From this, we see that case. Otherwise you got 1/3. And why center of Mass Watson? Try it off. That triangle is also look a to K away from the origin. So D location would also be to over three. They'll be known accordance off the central triangle. That is two or three than 203 soon Alice cock, Let m sub wine himself x less talk. Remember why we know that that is equal to mass times the distance from the origin. I mean on that mass is equal to the area. So let's start from the triangle. We have the triangle area as too. So we have two times one since the this says, um is one just to someone that is the area times negative 0.5. Now, this is the distance negative 0.5, um, plus area of the triangle, which is two times two divided by two months black by the ex central oId according to eccentric. Off that triangle, which is two or three. From this we see that I'm some wise that make up to 1/3 l s cock. I am setbacks area after trying rectangle is two attempts. One, uh, what is the y? Coordinate office central Read It is equal to one so two times one times one plus area after triangle, which is two times two divided by two. Well, it's black by y corn in it, which is two or three from this. Be fine and sub access tunnel three. Well, we know that X bar is equal to why moment divided by total less or, in this case, total area that is equal to won over three divided by total area here we founded us for, so that is equal to one over 12. Now why bar is equal to themself. Max give a total mess in this case, that is total area, and this is then equal to 10 over three D line. I foresaw that it's time over 12 or five or six. So that Sandrov mess or the Central Office whole system will be at one over 12 and five over six, okay?