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

$$\begin{array}{l}{\int_{C} y d x+z d y+x d z} \\ {C : x=\sqrt{t}, y=t, z=t^{2}, 1 \leqslant t \leqslant 4}\end{array}$$

$\frac{722}{15}$

Vector Calculus

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All right, so on this question were as to determine the line integral of life DX posi divide plus XTC over the curve. Given Parametric Lee with X equal spirit of tea life was easy was squared where t goes from 1 to 4. So the first thing we want to do is to convert everything in terms of teeth and DT. So, um, first we want to determine what the excess wealthy ex is just the derivative of square root of tea with respected t, which is just on over to the square root of tgt. And then the alliance just a derivative of why, with respect to so wise diva derivative of plywood keys Kiki. And finally, you see that your negative up to see where the respective t is? Just two tt. All right, So, um, so that should be He's all right. So the easiest to TV. All right, so now we convert. Now, we plug everything back to our original lining our original integral. So on re plug everything back. We get this magical right over here, and we can do some simplification. So, for example, T divided by square root of teams. Just t to the power of half. So we have 1/2 a t to the power. Plus he squared. And now tee times square root of tea is just tooted a power of one times two to the power of 1/2 which is just t to the power of one plus one. So one plus one is three halves. So we get to team did a power of 3/2 All right, now, this is just a bunch of, um, powers, which recounts that were adding just a bunch of keys to certain powers. So they're integral. It is pretty easy. We just about one to the exponents and then divide by the the number we have. So, for example, 1/2 plus one is three divided by two. So then we're gonna defined by three divided by two. That's equivalent to multiplying by 2/3 there, and we have t square, the integral Battisti Q divided by three. Then we have to now t to the power three over to the integral. Bad is to fifth chief the power off by paths, right? So now being determined are integral. We can simplify. So this half this out this to and to cancel in these to become a floor. And then now we're gonna take the integral with our parliament being foreign are lower limit being ones we're gonna plug in. Four. They were in a subtract looking one. So what we get is 7 30 70 divided by three What? 124 divided. But now the common denominator is we're gonna create a common denominator of 15. So you were gonna multiply by 5/5. Here we multiplied by three, divided by three. So we get 3 50 plus 372 divided by 15 which is just 722 divided by 15 that this is our valuable one.