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

$$\begin{array}{l}{\int_{C} x y z d s} \\ {C : x=2 \sin t, y=t, z=-2 \cos t, 0 \leqslant t \leqslant \pi}\end{array}$$

$\int_{C} x y z d s=\pi \sqrt{5}$

Vector Calculus

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so this question were asked to determine the lighting to grow. We're giving plan Tickle X Y Z theorist, and were given for and were given the para militarization of the Curve X is two scientist E Why is t c is negative two cups of tea and he goes from 0 to 2 pi. All right, so the first thing we have to do so our access given in terms of tea or rights given in terms of tea honors he's given here. However, DS, we need to determine it also in terms of tea. So we're gonna use the determined DS. We're gonna use this formula. It is a square root off the derivative of wrecks of respected. He's squared bustle, derivative of y with respect to the swearing plus the derivative of zero with respect to square on. Then we take the square root of that. I want to play a breaking all right now the derivative so access to assign t the derivative of that with respect to use to Costa derivative. Why, which is tea with respected T is just one the derivative of lysine which is negative two coastline to you with respected T It's just to sign. All right, so now we have to co scientist squared response. Where was two scientists Square? Well, to co sign t squared. Squared is four co signed square T two scientists were It is four signs square team. So we don't pull that four out as a common factor. And we're left with four times coastline, square size. And we know that close science purple sine squared to simply one. So we have the square root ball four times one plus one, which is a swear word of five on day duty. Direct. So DS is just a square root of five. All right, great. So now we have everything in terms of tease. We're gonna put everything back in, but then we have Now we have to deal with this into and our lord, Lord of the Middle Integral zero the over living despite. All right, what we can do is we can pull out a negative to square root of by, and we're gonna leave these two inside, and now we're going to see what so may have a negative to square with a five on the upside. Spare the integral from zero to pi of tea Time to sign He goes well to sign T coastline T This is a very things for your identity. This is just sign to t So this is just sign too. So now are integral becomes this race. So you have negative two square root of five times needed to grow from zero to pi off t signed to TVT And now again, this looks like a product. Uh, well, it looks like we have a product of functions inside the integral. So our first, uh, weaken used the integration by ports technique and we're gonna set Are you equal to t? So then you becomes DT on our TV is gonna be signed to TVT or the remaining, and so we becomes negative half co sign. Okay? Now, if we use the rule for integration by parts, which is you ve minus injury grow, do you? And then when we do that, we get this into growth rate. And then if with the Intergroup uh, negative 1/2 co signed through, teeth is just one over for sign too. So we have. So we get this into girl. So we saw the And then now we're just gonna plug in our over and Lord limits, which are applying zero. And then we're going to keep in mind that side of 00 Sign up to pipe zero. So really, over we're left with is negative 1/2. Hi. Hi. Co sign two parts. But we also don't co sign to Pious boy. So now we have negative two square root of fire on the outside. We also we also got a negative 1/2 pie on the inside. The Linus is cancelled. We have a 200 human greater doing the denominator. So we're just left with square root of five times place. That's our life.