WEBVTT
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in this problem were given such a shaped lemon.
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And where has to find experiment? Why moment?
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And the central mass off this whole system. Now
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, as we can see, this limiter composed off
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1/4 circle, 1/4 of a circle and a rectum
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. Now let's I doubt we're gonna perform to oppressions
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for each separately. Let's say that we're first interested
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in mess off the circle and we know that mess
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is equal to density Times volume. However, this
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is a lemon. It means that because in a
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thickness, so that'll be equal to row faster times
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the area circle D dust is given us six and
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this is a 1/4 of the circle, so the
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area would be pi times radius. Scribe radius is
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44 skirt for so d mess would be 24 pi
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unless had it. Now we're interested in mess off
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the rectangle. That would be the density which is
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again road times area rectangle. That would be six
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times area, which is the multiplication of the link
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of science. That I would be two times four
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and that would be 48. Now, what we're
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gonna do is this We're gonna find the scent right
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off direct angle and circle separately at first. And
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now we're gonna combine Does Since rectangle is a pretty
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ordinary shape, we know that central will be located
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right here. So cement TXs off direct angle.
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And we know that the mass off this rectangle as
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48 this would be the central a sense it is
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the 70 access the coordinates will be to then negative
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one. All right, we know everything that we
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need for the rectangle. And that was the easy
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part. Now let's work on the, um,
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best given circle. We would like to find sentry
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off this circle. Now we know that export foreign
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object, uh, is one over area integral x
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times D s. And why bar is you go
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to one or area internal. Why, Times D
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s now, let's find D s four d Circle
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. Ellis said it now at first were interested in
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X centrally. Okay, it means that we're taking
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10 strips tin sheets, which with off the eggs
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. And so the area off a tense trip would
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be why, times, dx And if you were
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to some those there years off, 13 strips up
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you would get the total area. However, the
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thing is, as the D exchanges as we move
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in, XX is d, why will change or
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to hide up that strip will change So you wouldn't
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need to use internal that what will be D d
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s well d s if we have such a tender
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vertical strip with a what off DX. It will
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be white and t x. And what is why
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, In terms of the ex, we know that
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this is a circle, so the question will be
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excreted. Plus y squared is a radius scapula radius
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is four settled before square it. So why,
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in terms of effects would Demby skirt of 16 by
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this expert team X All right, now let's head
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on. This is for export and four y bar
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You would have Yes, off circle Mrs Wyler's say
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would be except iwas. So the idea here now
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is that instead of having this vertical strips with with
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the X, we're having this for gentle strips and
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the wit now is d want. So then,
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uh, that would be exciting. D y what
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is x in terms of why I gotta reusing the
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same equation that'll be skirted of 16 minus y squared
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times D What? All right, now let's calculate
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the off those in trickles We have export as one
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or area where we already found the area off the
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one cardinal for circle here. So we have want
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over 24 pi inter go for X when we have
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our usual STX. As you can see, it
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changes from 0 to 4. So we have 04
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whatever 24 ply um, 0 to 4. We
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have X time scurried off 16 months X rayed the
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X Here, let's define a new variable Let's set
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out 16 minus X cried is our new Marie able
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you? So what are native to X? T
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X would be d'you and from this we can see
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that X d x will then be able to negative
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t you too. And, uh, doing that
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we now have a new interview for export form expires
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equal to well, actually, as we define you
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as 60 minus x courage, let's define limits for
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you when x zero, which is a lower value
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or lower limit for X. When X is here
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, you will be 16 and one x is for
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you will be zero. So now we have won
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over 24 piles. Inter go from 16 to 0
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. We have, um, skirt a few times
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. Negative. Do you? Or were too.
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And we can easily so this integral. And this
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will be to over three you to do three over
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two negative one over while I made a mistake here
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. 24 part here is a total mess. But
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the area will be only this part, which is
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four pies. So let me. He raised that
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So area is four pile. Okay, now we
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have negative one or eight Pine does. Are you
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changed between 16 0 from this p five x bar
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to be 16 or three pun And if he sat
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in jail, go for Why bar we have what
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over four point That is one over area again we
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have, uh except device I'd ever be Why Times
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started off 16 months. Y scribe, do I
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? As you can see, this is the exact
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same inter girl as this one. This should also
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be four pile okay. And the only difference is
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instead of X, we have one here, but
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the result will bit saying So while sent droids will
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be looked at you. Just 16 over. Three
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punch. Okay, now we know now the central
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. I'd order cortisol percent rate for 1/4 off a
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circle. Now let's work on that in a new
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page we have or 1/4 of a circle here and
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we have a rectangle now, you know, not
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District uncle has a central right here, and the
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co ordinates are too negative. One and now this
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1/4 on quarter per circle has the central. Somewhere
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in here in accordance are 16 over three pot and
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16 or three pie. We also know the mess
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off the circle. So M Circle is equal to
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24 piles, and M rectangle is a good to
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48. Now, let's think of it like this
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. Let's say that we had to point Mrs What
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guided out here and there. And we're trying to
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find the moment in what, and extractions We know
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that. And why is summation off mess Times Exposition
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Pseudo vegan meself rectangle With just 48 times the exposition
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, which is to plus on application off D mess
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off the circle. So 24 pond multiplied by the
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location off X century. So that is 16 over
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three pine Possible. Cancel out. We have eight
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here. So from this refund y moment off this
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whole system to be 2 24 Unless you say for
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X X moment will be summation off mess times.
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Why position? So the mess off rectangle is 48
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by position is negative. One plus mess off circle
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is 24 pi And why cornices 16 or three pine
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. These will cancel our what, eight. And
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from this we see that ex moment is 80.
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Now let's for exporting wild boar off this whole system
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. We know that export is equal to y moment
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, divided by total mess. That'll be then to
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24 divided by total mess as 48 plus 24 pi
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. And we can do at everything I ate and
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write this one as 28 over 28 over three times
149
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pi plus two. And why bar will be ex
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moment Delighted by total mess and that it will be
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80 delighted 48 plus 24 piles that is equal to
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the video. I ate again. 10 over three
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times. Pot plus two