WEBVTT
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So in this question we're giving the were given.
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You were asked to find the line integral of X
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squared D explosive voice, do I? And we're
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given that the curve conscious of two parts. First
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, it consists of the Ark of the Circle X
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squared plus vice, where people sport. And that
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goes from zero to you too. And the second
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part consists of the, uh, lying segment,
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which goes from zero to what if we draw this
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on a graph? So see, one just goes
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from zero, uh, zero too young to,
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And it's a circle of radius, too. So
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1/4 of a circle of radius, too. And
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then we're given the sea to simply a line segment
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. Okay, so says just buying second like this
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. All right, so first of all, we
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know how to pair of parameter rise circles, and
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we noted for amateurs like, So first, we're
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gonna start with C one. So are C one
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is on the quarter, the circle. So again
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, since it's a circle of radius for, uh
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, Well, when we write it when with a
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rabbit, arise it excess to close. I Inti
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lies to sign t and then our, uh are
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tee. Our TV goes from zero to pi over
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two. All right, so here it's clear.
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We started syrah. Go all the way up department
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. Okay, good. And then now, uh
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, we're gonna find the XO. The X is
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just the derivative of X, with respect it easy
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access to society. If we take the derivative of
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that with respected, you get negative to sign t
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and again, do you? Why the derivative of
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why with respected tea instead to coast 19 Right now
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, we're gonna plug everything back into the integral,
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and everything is gonna be in terms of tea.
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So are Lower LTD's from zero in our upper limited
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spy or two. And then we plug everything back
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. So place of expert enough to close 90 in
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place of why we're gonna two scientists in place of
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the expert gonna put negative to science t t in
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place of why we're gonna put do coastline. So
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we plug everything back. And then after we simplify
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, we get something, we get this right over
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. All right. Now, what can we pull
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out as a common factor? Well, there are
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multiple things we can float is a common factor.
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We're gonna pull out in eight, and then we're
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just going to switch. He's like this because this
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has a minus. I don't want The money is
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to be so Now are integral looks, Chris.
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All right, now let's seek This is a bit
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tricky because we're gonna have to do a double U
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substitution. So for the first part right over here
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, we're gonna set you equal sign, tear.
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And we have the EU is close 92. And
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for the second part, we're going to sit w
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equals two co society, and we get deep W
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is negative. Scientific et again, this is just
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a double use institutional tree already hard. Well,
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we should We should not tell right now. All
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right, so now we think the integral if you
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swear to you, that's just you cute, give
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you. And then we know that you in this
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side of you. So we get this treachery.
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And now the integral W Square. You tell your
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masters W Cube divided vice, and we know that
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w is co sign tease right over here. All
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right. And there's an eight on the outside.
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And remember, this is a definite integral. So
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we're going from pi over from zero to pi over
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two. Certainly plug this in by over two and
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zero and you subtract. We're going to get eight
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times 1/3 0 which is just one. Mine is
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your A plus 1/3 which is also so we have
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eight times 1/3 minus one 1/3 minus prefer to zero
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1,000,000,008 times zero, which is just okay. Now
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it's looking at C two. So see to we
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get the integral of Bill C to we have to
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parameter rise the line. So if we parameter rise
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the line, we know first, we need to
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find the direction in which the line is pulling.
83
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So we do X to marina sex sandwiches for minors
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, Europe is just four. And then why?
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Till minus final in which is three minus two,
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which is one. So then we get our parameter
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arised, uh, are parallel line is for so
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the line segment zero to the line for line segment
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connecting 0 to 43 has to be parallel to for
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all right, and then now we determine X and
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one So X is just explain, plus a teeth
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for X one is zero plus a is war.
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So zero plus 14 just 14 is our ex.
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Our Why is why one plus Bt's All I want
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is to and he is wrong. So it's two
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plus one team and our t goes from 0 to
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1. Why, with bunions Europe 14 get X
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equals zero. Why you two That's our first point
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. They plug in 1 40 we get X equals
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four y equals three, which is our second.
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And then again we're going to determine DX and dy
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y. The X is just a derivative before TV
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respected team just just 40 and the wise a derivative
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off two plus one to you and respected t just
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just tt okay, Now we plug everything back into
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the integral look everything and when we get everything in
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terms of but our limit points are from 0 to
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1. Okay, so now if we expand if
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we foil this right here and we multiply the side
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weekend 60 40 squared plus four plus support plus two
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squared. Now again, this we can combine the
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60 40 squared. With these squared, they get
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65 squares of get this integral over here, and
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this is not difficult. This is just the power
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room. So here we're just gonna integrates with the
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integral of 65 the spirit of 65 Cube, divided
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by three. Integral of forties to the square an
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angel for is just tea and are integral goes from
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01 So if we plug in one, we get
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65/3 plus two plus four minus of plug ins.
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0 40 Ridge is gonna get zero. So it's
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80. So now we do the we look right
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over here, and then we create a common denominator
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. We're going to get that the line integral is
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83 divided by three. Now, if we add
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a result for the first line integral for the quarter
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circle and then re at our second result which was
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the line integral for the line segment, we're gonna
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get zero plus 83 divided by three. It is
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just 83 divided