WEBVTT
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right. This is a problem. Number fifteen from
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section for chapter fifteen of early transcendence. A petition
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by James Stewart prompts us finds in your mass of
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Obama in the shape of the necessities, right?
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Triangle with equal sides of likely intensity in any form
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is proportional to the square of the distance from the
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Vertex opposite. I got news. It's the first
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thing we want to do is just strolling picture.
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So we're told that we have a right triangle,
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and each of the legs has, like, a
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all right. And so we're looking for the center
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of mass and the center mass is the point.
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Somewhere in this triangle, where is Jai? Balanced
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it on the end of the pen. It would
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not fall. It was stay perfectly balanced. So
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, in some sense, it's the average of me
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, Uh, X coordinate where all the masses,
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okay, and the white court, where all the
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mass is looking. So the first thing we're going
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to need to do is actually just compute the mass
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and so good. And so if we put in
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some sort of court system, so just put the
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X according exact sis along one bag in the play
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access number. We're gonna let this verdicts. The
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origin in there is our mill. It's protects me
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the origin because you know that the density is proportional
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to the distance from this pool and the origin is
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a great point to compute the distance from any of
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the nice distance formula from the origins. If I
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have this point right here, looks why and then
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the distance here, he's going to be square.
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Thanks for squaring because squared, that's the spirit sort
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of scream. Basically, just And so we're told
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the density of this triangle is proportional to this distance
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Where and so now if we wanted to find them
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ass of this triangle, the mass in general,
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it's from basic physical science is going to be the
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density times the area. And so if the dense
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New York constantly would just take that constant density vilified
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by the area is trying on your help, Max
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. Okay, but that's not kicks. If we
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take it's a little square here, Mom. Now
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, this small little differential square is goingto have area
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. Yeah, which is gonna be dx times.
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Why and then the small little square We can actually
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assume that the density is crossed. Okay? And
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so the density here, we're told, is proportional
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to the distance squared from the work. I talked
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to several of Dean. The intensity that any cortex
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, why it's proportional, which means there's some constant
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such that the densities that constant times square y squared
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and now knows that we don't know this number's seek
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. So either we don't have enough information. So
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the problem or is going to turn out to not
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matter and that's going to be the case. The
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center mass is going to be independent. That's proportionality
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constant. So in our square over here, we
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can assume that the density is constantly equal to this
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value given by the ex Michael. And so we
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have the small, little different mass cam. This
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is just Sydney the point only sometimes. Thanks.
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It's just good, Erica. And now we want
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to know the total mass. You can just add
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up all these little squares. And now, of
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course, these air differentially small squares. So we're
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gonna have to use a vinaigrette. And because we're
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looking for two dimensions ex, um, I never
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actually he's a double emigrant. And so what?
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We have is a mass the girl for X and
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y Well, I am actually could end that you're
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sort of you're gonna have to get X and y
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involved so we'Ll have to put in our limits of
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integration. Why? But the grand if something see
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x squared square Yeah Excuse me Now the way I
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have written I've written the ex first So we'LL do
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this eccentric girl first actually doesn't matter by fifteen East
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from this anagram is kicking a differential so we can
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actually embrace more want But if we look at X
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first we're going to fix a why a white coat
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. So here and now, where is X going
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excess starting at zero? Go away until you get
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this line here. And what is this long line
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when you're right? This is a function of why
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? Well, it's not too hard to see that
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it's going to be okay. Us What? There's
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a couple ways to see that It's a slope.
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It's a line of slope negative one as X intercept
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a year. But that's what it's going to be
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. So X is going to go from zero Tuesday
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. Why and why is gonna be going from Europe
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more? No, you actually compute. Yes.
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And I know it seems that it doesn't have anything
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to do with Senator Vast. But we really do
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need to know the master calculated the center of mass
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because, well, the similar mass is going to
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be basically an average over this mess, Senator,
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but versus just compute this shit. Like I said
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mass giving about this on your girl going from you
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, eh? Thanks, guys. Hey, why
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in your hand? So we have this constant.
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Just get affected that out of the thunder girls,
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then we have square by square. Why now?
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We just gotta work. So we're doing x first
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. So we need to find an anti guerrilla of
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dysfunction, huh? Just a polynomial. So anti
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drug X squared X cubed over three. Why squared
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? It's constant from the excitement, which is sex
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by squaring. They're evaluated zero minus. Why?
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It's playing in a mine. Why? Yeah,
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it's over. Hey. Hey, Mike. Why
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? You know, So I really don't need to
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plug in the lower limit because it's your functions here
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. Okay, Now, this is just a polynomial
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. Consider a couple ways to do it. You
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do. U substitution No, or you can just
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expanded out. I prefer just to expect it out
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. Um, they really doesn't matter. I mean
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, this first one, if you let new a
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Miles, why, you? Well, we're just
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going to get a negative sign here. Why?
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And so this first, then a girl. I
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was just going to He's my everything. So I
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can just look at this first time right here.
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Do this substitution. It's just going to me,
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the inner girl. Okay, So why is equal
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to zero use from a twice a zero, You
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three, near where we have this negative sign,
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King and bless this one. I'm just going to
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expand out. Okay, Clyde. Great minds,
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human. Bye. Okay, so one thing we
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can do right away as we can just get rid
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of this negative sign if we just switch and zero
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. And at this point, this is really just
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a count. One question. So this this integral
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here. Well, when I take the anti Drew
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that you can't, you're going to get you to
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the fourth of four, which is going to give
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me a U to the fourth of twelve. And
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then I'm gonna value zero and about you today.
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That's going to give me minus a to the force
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over twelve minus zero. But that whole thing zero
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so begin. I'm just kidding. Four acto from
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this guy and then plus take Nancy Drew here.
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Hey, why? Yeah, And miss line Where
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? Four. I went from zero to you.
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Okay, but and I'm just going to get subtraction
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. Look at together. What ain't it forth?
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Twelve? His term comes with a painter before over
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three minus into the forest for, you know,
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I'm almost done. So now we just have to
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add. Okay, so here's one You know,
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the fourth number twelve. We had a common denominator
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. This's going to be for need to the fourth
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of it all, Linus Missile. Me three needed
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the fourth number twelve. So all in all we
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get to you. Still other cops in there,
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too. And, of course, twelve Cher's C
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times. No, over six. And so that'LL
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be our mass. And now notice our masked depends
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on this constant C and it depends on basically the
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size of a triangle. We're not that That's just
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the mask. So what is the mass have to
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do with the senator's glass. Well, if we
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look back at this triangle, we found the mass
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. You know what? The X cordon of December
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Mass that's gonna be here long. This leg,
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What I could D'Oh, they just cut up the
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rectangle in strips and I just want to take and
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each one of these Canada and infinitely, uh,
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thin strips. Say us one. Well, I
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want to take the mass and waited of this Valued
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X in the god of all those those massive trying
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his X coordinates Well, then divided by the total
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mass. And that's just going to give me an
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average of for the masses along the X axis.
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And I want to take it of course room You're
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a over the hole triangle. So basically, what
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I'm saying is I want to integrate, add up
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all of the mass, but I want await it
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with this X, and that's gonna give me kind
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of expected value. You know where the mass is
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and then and just getting by. It's the same
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thing. Where what? I just want to add
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up all the mass. It's raining. It's why
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, says Wise is telling me where alone it's war
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horizontal strips. Kind of what? The masses that
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each one of those strips. And then that'LL just
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give me an average. That I will. Then
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what mask? This is very much analogous if you're
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confused too, you know, finding an average grade
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. What am I doing in the numerator? I'm
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just adding up, uh, kind of, you
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know, the x coordinate. This is like a
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throne of students in the class, the total mass
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and each one of those strips for your students that
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their grade and then I just dividing by the total
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number of students, something like it really is just
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an average. And again, wherever this point is
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, that's going to be the balancing point. Also
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called the Essentially Hey. Okay, So I really
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need to do is just compute news to dinner girls
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here and we'LL start with ex. So if I
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look at this girl Thanks. Okay. Well,
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actually, your rights to recall what d mm.
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Is this the sea experience wise? Great. Why
206
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? And then of course, the same integration will
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be the same. So it's more there. So
208
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your call. We're going from zero to aimless.
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00:17:47.051 --> 00:17:59.112 A:middle L:90%
Why? What is going on here? A and
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I'm waiting with X here, X woman tm it's
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going to be okay with this this constant understand?
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Good impactor out because you've been extra. Why?
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00:18:15.604 --> 00:18:18.663 A:middle L:90%
And I have this sort of Vincent me escort y
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squared in the meantime study area. Thanks, Buck
215
00:18:29.703 --> 00:18:30.364 A:middle L:90%
. Well, actually, I think this one is
216
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always a little bit easier, at least initially.
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So we do. The X and a girl first
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actually have this X here, which is going to
219
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allow me to do a nice the substitution of Yuki
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X squared plus y squared. Yeah, thanks.
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Yes. No. So then I can replace this
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x Times GX in here. You So get this
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factor in half that from California's just good practice to
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put all the constants out of the immigrants. Defector
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. Thank you. No, Mark it. So
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I have sex. The ex gets dissolved into you
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, and I just have a new here still monkeying
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while here. Nice sniffing out where my unit is
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00:19:45.273 --> 00:19:48.374 A:middle L:90%
going, OK? Selling why is equal to that's
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00:19:48.374 --> 00:19:55.554 A:middle L:90%
where X is equal to zero square. Yeah,
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00:19:56.523 --> 00:20:02.074 A:middle L:90%
right. And then what? X is equal to
232
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a mine? Why? Well, I have a
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minus y squared. So here in excellence. Maybe
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minus y. Here. What? Scream? Great
235
00:20:23.913 --> 00:20:34.644 A:middle L:90%
. Hey, hey. Why? Why? Good
236
00:20:37.983 --> 00:20:42.794 A:middle L:90%
, Great. And so just actually do that.
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00:20:44.054 --> 00:20:55.354 A:middle L:90%
CEO Yoo, No, to a right now.
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This is going to become you squared. Yeah.
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00:21:03.523 --> 00:21:14.094 A:middle L:90%
Square. Never again about getting this from black squared
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00:21:14.263 --> 00:21:22.114 A:middle L:90%
, Tio. It's Mars plus screaming. Okay.
241
00:21:23.354 --> 00:21:27.584 A:middle L:90%
All right. So you can factor out this factor
242
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of from what to see for now. Okay,
243
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So, integrating a right? Yeah. Big jacket
244
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. First term is news. A month squared,
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squared, squared minus. All right. Good for
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00:22:07.219 --> 00:22:14.378 A:middle L:90%
you. Oh, right. So it's not the
247
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nicest thing, but it's doable. All right.
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00:22:18.719 --> 00:22:23.298 A:middle L:90%
So again, we could expand this out. Uh
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, that's probably the best thing to do with this
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point. So if we have thank you. No
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, I didn't mean to do that, Thank God
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. Okay, so I have to live back,
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so All right, See, Four. It's good
254
00:23:06.269 --> 00:23:10.979 A:middle L:90%
. Right this way. And, Grant, I
255
00:23:10.979 --> 00:23:14.929 A:middle L:90%
was ever going to square this. Are you a
256
00:23:14.959 --> 00:23:18.878 A:middle L:90%
A minus y squared? It's wise, friend.
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So nice way to just distribute about some squares.
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Four. Can you really have two times squared?
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00:23:40.169 --> 00:23:56.098 A:middle L:90%
A minus y squared spider and minus what course?
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So you and something good happens, they're wise to
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00:24:07.236 --> 00:24:11.705 A:middle L:90%
the force will cancel. Yeah, tens over it
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is like this. And so this first guy,
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we definitely just wanting the substitution, so just let
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me be. That's why before you can even do
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it. So we're just going to pick up a
266
00:24:34.625 --> 00:24:48.105 A:middle L:90%
negative sign? Come on, go ahead and skip
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a couple steps. Essences Kwan. So there's a
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00:24:52.306 --> 00:24:56.076 A:middle L:90%
Miles one of the fourth is is if I just
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00:24:56.076 --> 00:24:57.076 A:middle L:90%
do the substitution, I'm going to get a negative
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00:24:57.576 --> 00:25:00.316 A:middle L:90%
, But it's just gonna be a minus one.
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But of course, with these negative, so eh
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Okay, well, then, next turn. Well
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, let's just expanded out, and then we're gonna
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have some scratch. Very quite scared, eh?
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00:25:34.695 --> 00:25:42.855 A:middle L:90%
No. Hey, s So we should probably go
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00:25:42.855 --> 00:25:45.236 A:middle L:90%
ahead and expand this out. Ms. Was going
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00:25:45.236 --> 00:25:52.986 A:middle L:90%
to be what? Squared Squared? Minus two.
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00:25:55.425 --> 00:26:02.796 A:middle L:90%
Why? Cute. What? Or all times?
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This too. That's what we're immigrating and a great
280
00:26:10.915 --> 00:26:19.006 A:middle L:90%
first term. Just get why schemed a squared over
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00:26:19.996 --> 00:26:37.965 A:middle L:90%
three minus Teo. Why? Over. Okay,
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00:26:38.316 --> 00:26:45.806 A:middle L:90%
and that's it. Soon our kids see over for
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00:26:48.086 --> 00:26:51.506 A:middle L:90%
and now when I go out with a this first
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chairman, get zero on the negative skin cancer with
285
00:26:55.346 --> 00:27:03.425 A:middle L:90%
the negative, I subtract five. Oh, right
286
00:27:03.496 --> 00:27:10.405 A:middle L:90%
. Plus Okay, Teo, And look a here
287
00:27:11.246 --> 00:27:18.375 A:middle L:90%
. They're going to get in a three. Hey
288
00:27:21.796 --> 00:27:27.066 A:middle L:90%
, keeper for. But we also have this factor
289
00:27:27.076 --> 00:27:32.056 A:middle L:90%
to move to a little bit more in class.
290
00:27:33.895 --> 00:27:44.405 A:middle L:90%
Who now? It's just more had infractions are here
291
00:27:44.996 --> 00:27:52.046 A:middle L:90%
. Reform. Five common nominator. Uh, well
292
00:27:52.195 --> 00:27:53.705 A:middle L:90%
, it looks like we're gonna have to get over
293
00:27:53.705 --> 00:28:06.705 A:middle L:90%
yet, too. Snakes. Hey, and go
294
00:28:06.705 --> 00:28:11.236 A:middle L:90%
on. So here we have twenty. And of
295
00:28:11.236 --> 00:28:15.046 A:middle L:90%
the fifteen or sixteen, and now here, you
296
00:28:15.046 --> 00:28:22.215 A:middle L:90%
really have to service Will be Ernie End of the
297
00:28:22.226 --> 00:28:27.635 A:middle L:90%
Fist sixteen. And here we have what that's going
298
00:28:27.645 --> 00:28:37.715 A:middle L:90%
to be. Well, well and then a questionable
299
00:28:38.405 --> 00:28:41.336 A:middle L:90%
too. So it looks like we're getting too.
300
00:28:41.336 --> 00:28:55.846 A:middle L:90%
So I totally get it for over six. Look
301
00:28:56.385 --> 00:28:57.766 A:middle L:90%
inside now. Eight of the fifth over five.
302
00:28:57.766 --> 00:29:03.076 A:middle L:90%
Again. That's twelve. And the fifth over sixteen
303
00:29:03.086 --> 00:29:11.506 A:middle L:90%
. Plus for here. Twelve others, of course
304
00:29:11.885 --> 00:29:17.816 A:middle L:90%
. Yeah, for see, That's going to be
305
00:29:17.826 --> 00:29:25.526 A:middle L:90%
sixteen. So get sixteen. Can I see you
306
00:29:25.536 --> 00:29:33.586 A:middle L:90%
for a course? Sixteen. Sixty More than a
307
00:29:33.596 --> 00:29:41.605 A:middle L:90%
simple prize, too. Sync the fifth over.
308
00:29:45.596 --> 00:29:48.536 A:middle L:90%
All right, Snow Done. Kelly's with the X
309
00:29:48.536 --> 00:29:55.905 A:middle L:90%
coordinate. The X coordinate mass call is this double
310
00:29:55.905 --> 00:30:02.215 A:middle L:90%
and a girl. Thanks. Um, a mass
311
00:30:04.135 --> 00:30:15.655 A:middle L:90%
, actually. Hey, team sing. Hey,
312
00:30:18.536 --> 00:30:22.546 A:middle L:90%
six, You call and this is going to simplify
313
00:30:22.605 --> 00:30:27.915 A:middle L:90%
to a over five. No notice seized. You
314
00:30:27.925 --> 00:30:30.766 A:middle L:90%
canceled to the senator. Masses of people see.
315
00:30:30.766 --> 00:30:33.895 A:middle L:90%
But of course, it does depend on the size
316
00:30:33.185 --> 00:30:37.786 A:middle L:90%
of the triangle. And now we could do exactly
317
00:30:37.786 --> 00:30:38.996 A:middle L:90%
the same thing For the Y center of mass,
318
00:30:40.635 --> 00:30:42.836 A:middle L:90%
you can get in the exact same wonder girl.
319
00:30:42.836 --> 00:30:47.385 A:middle L:90%
It would work exactly the same. But something really
320
00:30:47.385 --> 00:30:51.526 A:middle L:90%
nice to notice is because of the symmetry of the
321
00:30:51.526 --> 00:30:55.566 A:middle L:90%
density. If isis switch at X and why nothing's
322
00:30:55.566 --> 00:30:56.115 A:middle L:90%
going to change. I mean, you can also
323
00:30:56.115 --> 00:31:00.175 A:middle L:90%
see that when we actually do this Computer wasn't a
324
00:31:00.175 --> 00:31:03.796 A:middle L:90%
girl. If I just replace this with why I'm
325
00:31:03.796 --> 00:31:06.306 A:middle L:90%
getting exactly the same, you know, grand,
326
00:31:06.306 --> 00:31:08.425 A:middle L:90%
just relabeled with X and y said the Y center
327
00:31:08.425 --> 00:31:11.066 A:middle L:90%
mass. It's going to be the same today.
328
00:31:12.205 --> 00:31:15.645 A:middle L:90%
But if you don't believe me, of course you
329
00:31:15.645 --> 00:31:18.246 A:middle L:90%
can just damage all. It's very similar