WEBVTT
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So this question were given that we have a thin
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wire and it has this shape of Sandy's surf semi
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circle of radius R. So I'm were given that
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the Ben City Oh, the density function is equal
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to K X one, and they were asked to
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determine the mass in the center mess. All right
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, so the first thing we're gonna do we're gonna
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parameter rise this this current. So since we're given
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that it's a certain part of her circle, we
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know that our X and part of the Circle Freddy's
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Bar of Radius eight we know that X is going
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to be a co sign t. And why is
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gonna be a society? That's how we correct tries
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a circle and it goes from zero to privatise tears
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between zero and five or galicians your own apartment.
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All right. Now, to determine the mass,
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the mass is just divine into wrote brought off the
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density function. I know we want to write everything
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in terms of keep so first followed. Yes.
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So, first of all, we know that we
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can write row in terms of TV because X is
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in terms of tea and wines in terms of tea
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. So that's simple. I know you're not.
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We want to write the yes, in terms of
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TG or GT. So we know the formula,
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sir, The yes is the square root of the
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X by people, I swear what people are like
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that sorry DX by DT swear plus divided squared.
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So now the derivative of EKO sank A with respect
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to t is negative. A sign, T and
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interpretive of plywood perspective teas. Just a closer no
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, only square those we get a sward which we
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can follow as a common factor. And what we're
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left with is, um, co sine squared plus
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science Berg, which is one. So we get
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that, I guess, is just a The square
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is a square root on this were cancelled left.
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So that's DS. So now we're gonna put this
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back Our mass function our mass. So m is
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square root from zero to buy over to k a
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co sign t a sign T e d t.
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Um, what we can do is we're gonna unplug
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the constants which are a que Pincay. But now
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we're gonna do something that might know you clear it
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first. So we're gonna divide by two and multiply
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so just this like a sneaky one. So I
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pulled out 1/2 so I'm gonna multiply. What's inciting
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to grow by two. All right, now what
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do we know? Where? Trigger identities. We
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know that to close on 90 scientists just to sign
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up to. And now what's Indians? But we
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can stop the internal that we have. We can
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solve using you substitution. Zor gonna get you equal
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to t So do you is just to DT.
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And if we divide by 21 of both sides,
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we get the details. Justin, you divide effective
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. So now we determined our DT, which is
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just do you divide it by two. And now
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this integral there's really and just one last thing explosions
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. The tea goes from zero to high over to
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are you is gonna go from zero because we're gonna
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multiply zero by two. This is zero. And
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then, instead of ending it to ah, over
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to you is gonna end up time because two times
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So we're using this relation right here. So two
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times 5/2 is just so that's why our limits right
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now we can pull, but half to be outside
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. You know, we're left with big cube times
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k divided by four and the integral of sine of
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you d years. Just negative co sign for and
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are integral are limits are zero. All right,
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so now we put high and zero in Do co
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sign. So we get, uh, a cute
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K divided by four times negative. Negative one plus
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ones of this whole thing right here. It's two
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and two and the four simplify. So I left
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with a cube times Katie about it, too.
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So that's ordinance. All right. And now we
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need to determine the center of mass of center of
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massive pecs we're gonna name it. Accepts e is
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just one divided by the mass and the integral of
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X raw. Devious. So we know that I
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am Is a cube k divided by two. So
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do one divided by that, and then we're gonna
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plug in now. Everything else become great in terms
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of, so we just park that in, and
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then we're gonna get we're gonna pull out the constants
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which Peter the four times K. And here we
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can simplify cubes. And either the power forward just
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left with a case canceled or just left with chili
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. And now we have course. France were signed
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TVT Well, this one again. We can use
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U substitution. Zoff said you equal to pro sign
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TV that you is negative. Sign tgt, Which
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means that scientist DT is negative to use of this
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whole thing right here is just negative. So now
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we get this into girls right over here. All
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right. Now, what's the integral all, um
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, of negative? You swear to you that's just
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negative, You cube you. But are you escrow
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sine of t? So that's how we get this
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integral right here. And then Now we just poured
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in the limits, which is part of her two
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and zero. So we get to a times zero
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lost 1/3 or to be divided by three. That's
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our X coordinate. Now we need to find our
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bike work again. This is exactly the same steps
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as before, except instead of having X, we
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have one. So we're gonna follow exactly the same
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steps as before. We're gonna pull up Constance and
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so on. But the only difference now is that
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the integral right inside here is a bit different in
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the case above, we had co signed squared scientist
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here we have sine squared closely. So instead of
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taking you to be co sign, we use u
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substitution, we let you be equals B with you
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equal sign. So the you just close on TV
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. All right, so now we go listen to
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over here. All right. Now, what's the
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integral of your square to you? Well, that's
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just you, Cube. You, but we know
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that you assigned t so this is just signed Cube
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t divided by All right. So the integral a
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few square to use you, Cube, divide three
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. But are you? It's signed. So now
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we get this in general, right here and that
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. Sorry. Now we get this function right here
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. This question and what we do is we know
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, plug in the limit. All right, So
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sign of pile or two is just one. So
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we get to a times 1/3 minus sign of 00
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But workers on zero, which is just to a
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divided by. So now we determined both coordinates for
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the center of mess, which is to a derived
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by three on 2 to 8, divided by three
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. So our center master coordinate Is this right here
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? This spirit to a divided by three common to