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Okay, folks. So in this video, we
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have this line into grown. That we need to
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evaluate. This is this is problem number 11.
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We have this London grow and to grow along the
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curve C of X y plus why, plus Z
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Uh, yes. Okay. And were given x
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and y and see all of them as a function
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of t so excise the function of T is to
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t. Why, as a function of t is
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t and Z is to minus two t. Okay
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, So the way we're gonna do this is the
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usual way of evaluating line into girls which is by
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converting DS into something times DT. And if you
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remember and DS is just ex prime squared place.
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Why Prime Squared plus z prime squared The whole thing
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multiplied by DT and ex prime white Bar Mzee Prime
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, of course, are all just notation for a
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DX DT and D y d t indeed CDT.
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So what is the x DT? Well, the
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expertise to because X is to teach the dx tts
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you no two. So two squared plus the same
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thing with why but wise teas. And we have
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one squared plus minus two squared multiplied by D t
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. Now that's equal to four plus four plus one
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plus four, which is nice d t but nine
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. The square root of nine is just three.
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I hope you know that. So DS is three
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d t When a plant this back into the line
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integral. So we have the integral if along the
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curve C of x times What? What's x times
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? Y well x times Why is this thing times
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this thing? So that's those two t squared right
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to t squared. Plus what's why? What?
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Why is t and then added into Z? But
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he is two minus duty. Okay, that's nice
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. The S is three d t. I'm gonna
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pull the constant out in front. Now let's evaluate
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the line integral. So we have three ah to
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t squared minus two t plus ts minus t two
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plus two. Multiply it by DT. So now
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it's Let's evaluate this this integral here. So we
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have three of, um to over three t three
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minus one Hertie squared plus two t evaluated at the
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two other to our limits of integration. The 1st
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1 to 0 and the top one is one.
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Let's do it. Let's plug in and let some
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do some plug and chug. Here we have three
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. I'm the, you know, plugging the one
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and zero, respectively. We have two of her
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, three minus 1/2 plus two. That's for the
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top limit minus 000 which is just zero. So
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this is gone. So we have, um let's
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multiply the three inside of the parentheses. So we
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have two minus three have plus two. Oh,
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excuse me. Not plus two plus two times three
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plus six. Okay, so So this is eight
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minus 3/2 which is, uh, a fine at
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1.5, which is Ah, six point fire,
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which is 13 over to. This is the answer
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for this video. Thank you, but I