Calculate the moments $ M_x $ and $ M_y $ and the center of mass of a lamina with the given density and shape.
$ \rho = 6 $
Applications of Integration
in this problem were given such a shaped lemon. And where has to find experiment? Why moment? And the central mass off this whole system. Now, as we can see, this limiter composed off 1/4 circle, 1/4 of a circle and a rectum. Now let's I doubt we're gonna perform to oppressions for each separately. Let's say that we're first interested in mess off the circle and we know that mess is equal to density Times volume. However, this is a lemon. It means that because in a thickness, so that'll be equal to row faster times the area circle D dust is given us six and this is a 1/4 of the circle, so the area would be pi times radius. Scribe radius is 44 skirt for so d mess would be 24 pi unless had it. Now we're interested in mess off the rectangle. That would be the density which is again road times area rectangle. That would be six times area, which is the multiplication of the link of science. That I would be two times four and that would be 48. Now, what we're gonna do is this We're gonna find the scent right off direct angle and circle separately at first. And now we're gonna combine Does Since rectangle is a pretty ordinary shape, we know that central will be located right here. So cement TXs off direct angle. And we know that the mass off this rectangle as 48 this would be the central a sense it is the 70 access the coordinates will be to then negative one. All right, we know everything that we need for the rectangle. And that was the easy part. Now let's work on the, um, best given circle. We would like to find sentry off this circle. Now we know that export foreign object, uh, is one over area integral x times D s. And why bar is you go to one or area internal. Why, Times D s now, let's find D s four d Circle. Ellis said it now at first were interested in X centrally. Okay, it means that we're taking 10 strips tin sheets, which with off the eggs. And so the area off a tense trip would be why, times, dx And if you were to some those there years off, 13 strips up you would get the total area. However, the thing is, as the D exchanges as we move in, XX is d, why will change or to hide up that strip will change So you wouldn't need to use internal that what will be D d s well d s if we have such a tender vertical strip with a what off DX. It will be white and t x. And what is why, In terms of the ex, we know that this is a circle, so the question will be excreted. Plus y squared is a radius scapula radius is four settled before square it. So why, in terms of effects would Demby skirt of 16 by this expert team X All right, now let's head on. This is for export and four y bar You would have Yes, off circle Mrs Wyler's say would be except iwas. So the idea here now is that instead of having this vertical strips with with the X, we're having this for gentle strips and the wit now is d want. So then, uh, that would be exciting. D y what is x in terms of why I gotta reusing the same equation that'll be skirted of 16 minus y squared times D What? All right, now let's calculate the off those in trickles We have export as one or area where we already found the area off the one cardinal for circle here. So we have want over 24 pi inter go for X when we have our usual STX. As you can see, it changes from 0 to 4. So we have 04 whatever 24 ply um, 0 to 4. We have X time scurried off 16 months X rayed the X Here, let's define a new variable Let's set out 16 minus X cried is our new Marie able you? So what are native to X? T X would be d'you and from this we can see that X d x will then be able to negative t you too. And, uh, doing that we now have a new interview for export form expires equal to well, actually, as we define you as 60 minus x courage, let's define limits for you when x zero, which is a lower value or lower limit for X. When X is here, you will be 16 and one x is for you will be zero. So now we have won over 24 piles. Inter go from 16 to 0. We have, um, skirt a few times. Negative. Do you? Or were too. And we can easily so this integral. And this will be to over three you to do three over two negative one over while I made a mistake here. 24 part here is a total mess. But the area will be only this part, which is four pies. So let me. He raised that So area is four pile. Okay, now we have negative one or eight Pine does. Are you changed between 16 0 from this p five x bar to be 16 or three pun And if he sat in jail, go for Why bar we have what over four point That is one over area again we have, uh except device I'd ever be Why Times started off 16 months. Y scribe, do I? As you can see, this is the exact same inter girl as this one. This should also be four pile okay. And the only difference is instead of X, we have one here, but the result will bit saying So while sent droids will be looked at you. Just 16 over. Three punch. Okay, now we know now the central. I'd order cortisol percent rate for 1/4 off a circle. Now let's work on that in a new page we have or 1/4 of a circle here and we have a rectangle now, you know, not District uncle has a central right here, and the co ordinates are too negative. One and now this 1/4 on quarter per circle has the central. Somewhere in here in accordance are 16 over three pot and 16 or three pie. We also know the mess off the circle. So M Circle is equal to 24 piles, and M rectangle is a good to 48. Now, let's think of it like this. Let's say that we had to point Mrs What guided out here and there. And we're trying to find the moment in what, and extractions We know that. And why is summation off mess Times Exposition Pseudo vegan meself rectangle With just 48 times the exposition, which is to plus on application off D mess off the circle. So 24 pond multiplied by the location off X century. So that is 16 over three pine Possible. Cancel out. We have eight here. So from this refund y moment off this whole system to be 2 24 Unless you say for X X moment will be summation off mess times. Why position? So the mess off rectangle is 48 by position is negative. One plus mess off circle is 24 pi And why cornices 16 or three pine. These will cancel our what, eight. And from this we see that ex moment is 80. Now let's for exporting wild boar off this whole system. We know that export is equal to y moment, divided by total mess. That'll be then to 24 divided by total mess as 48 plus 24 pi. And we can do at everything I ate and write this one as 28 over 28 over three times pi plus two. And why bar will be ex moment Delighted by total mess and that it will be 80 delighted 48 plus 24 piles that is equal to the video. I ate again. 10 over three times. Pot plus two