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Problem 31 Hard Difficulty

Find the exact value of $ \int_C x^3y^2z \, ds $, where $ C $ is the curve with parametric equations $ x = e^{-t} \cos 4t $, $ y = e^{-t} \sin 4t $, $ z = e^{-t} $, $ 0 \leqslant t \leqslant 2 \pi $.

Answer

$\frac{1}{8}-\frac{1}{8} \cos (16 t)$

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Video Transcript

Okay, so I'm going to explain the setup of this problem. So I'm not completely sure. Author, house author, design department Who is solvable? I think I'll stop. It's the cinema and explain why First of all that the way to this kind of blind, The whole part of the first things to fill out Yes. So we computed derivative with respect to tee off home Finish off that Megan he appealed by for that group Chairul Thank you. Yes. We'Ll be at this after you simply for using the fact that Science Corp Roscoe's and spirits What? You get this And, uh so that's trying to figure out what was it A crow is We tried to integrate from zero to two parts. Ask you so is heat with a negative three t Oh, Sy Q forty. Why square it'll negative to Teesside Square forty z The s is three squared off too Not funny e t So we have everything set up zero to two pi We put three squared off to you for months and we have each with a nephew seventy coz i cube for teeth Sai square for teeth Hey, so this will be a very difficult. Integral to it. I don't think I'll completed because I have to use our other five different pages, too, to finish it. Let's give you an idea how to how to deal with qualities like this. Well, maybe the things you can do is visit ideas in general when you're dealing with trigonometry, integral high power are usually harder is with the always They're larger and goes, So the idea is you use product to some former to to to write the product of true gumption function into the sun and, as you know it shop the exponential function times trigonometry function without the power is some power in your manage both to Dubai. Use the integration by parts they're being say, still very lengthy process so I can try to show you how to deal with is the first step. For example, co sign square forty size square forty. This will be one over to sigh eighty square. So one over for size square eighty, this will be one over eight minus one over eight. Co sign sixteen t. Yeah, and, uh, but that's not everything we have. We have coast. We have another co sign. Fourth he So we have. So this part's everything I'm writing. Red should be one over eight forty minus one over eight. Co signed sixteen p. Coz I'm forty here. We have to use another product of some form lock in. So coz I off our coast on beta is, uh is co sign off a plus, baby, Huh? Minus coz I off minus pate are over two. We have to divide this. Those are here, I think is plus no anyway. And did you hear? So anyway, we covered it into this and it's off them. We can do the each of them, like, eat like that If eighty co sign Bt this all each of them is the standard power in your congress too. Where you do the introversion by part and do the rearrangement to you'LL be able to, uh, together value off this integral and as you and you have the pocket in here which I think is way too long for the purpose ofthe one video. But I hope I hope this this explains the concepts. And if you want to go on and on, figure out the detail you can You can't do it yourself. Or if you want to just know the approximate value you can use a computer to compute this