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Evaluate the line integral, where $C$ is the given curve.

$$\int_{C} x y d s, \quad C : x=t^{2}, y=2 t, 0 \leqslant t \leqslant 1$$

$\int_{C} x y d s=\frac{8(\sqrt{2}+1)}{15} \approx 1.2876$

Vector Calculus

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So if this question were passed to determine the lining to grow Uh, X Y yes. And we're given that the curve. We have a pyramid to repair amateur eyes curve where X is equal squared and wise. He would take two to t and we're told that goes from zero 21 All right, so the first thing we're gonna do is we're going to determine or clank for DS. Then we know that the S is square Heard of d y by d Squared plus DX by DT square TT So if we take why and we finest derivative with respect it about activity, so lies just to t so that the derivative of, uh, the derivative of that with respect to, uh, t so to t the derivative of to tea with respect a tease just to And then also we take a riveting vax with respect. Cities of excess just t square. The derivative of that with respected tea is simply too All right. So finally you get square root of two squared plus two. So this just four plus 40 squared. We can pull out of forest common factor the square root of forest to And then what's left inside this square and square root is just X squared. So we determined our ts, our ideas we have we know that access t squared. We know that. Why is two teeth so simply what we're gonna do is we're gonna plunk now everything back into this and try to determine. Okay, so now when we plug everything back, we get this big mess right here, but we're gonna try that still, So we pull out the tour. So two times two is four so we can pull that out as a constant And what we're left with tears. That's right. So we haven't too sweaty square, plus one under the square root, and we have a cubed on the outside. Well, let's try to solve this integral using substitution. So we're gonna sit you equal to the square, plus work, and then if we take the derivative of there we get to you is to teach. And then if we divide by two on both sides, we get that TV team is equal to you, divided by two. Also, we're gonna do something that might be a bit unusual, but, you know, we can also since we said u equals C squared plus one. What we can do is we noticed that he squared. Is he good to you? Minus one. So no, this is create because that we can write everything in the integral here in terms of you. So also, we were gonna change the limits so the lower limit was zero, but that we know that there's a mystical tissue we can pluck that into t squared +10 square plus one is just one. So that's our new limit. Because we want everything in terms of you and again, we have our upper limit off. The integral is teasing cool toe one. So if we plug that into use equal to the square Plus when we get one square, plus one more to So now we completely then we have everything in terms of you. So everything we have in terms of you and R d t is just Do you divided by two so we could pull that tutor the outside So four divided by two is just all right. So now we have to take the integral, uh U to the power of 3/2 minus you did well. The integral beauty, the power of three divided by two, is just due to the power 5/2 times Tool and the integral of U to the power half is disputed. The power 3/2 times two of us. And the limit is from 2 to 1 of these. So now we just plug everything back in. We don't forget we have a two on the outside. But then if we work out the out abruptly put this into our calculator, we're gonna get that. Our final answer is 1.2876 So that's the value of the line.