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Problem 36 Hard Difficulty

Find the mass and center of mass of a wire in the shape of the helix $ x = t $, $ y = \cos t $, $ z = \sin t $, $ 0 \leqslant t \leqslant 2\pi $, if the density at any point is equal to the square of the distance from the origin.

Answer

$\frac{-4 \pi}{8 \pi^{2/3}+2}$

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Video Transcript

use a computer, the center ofthe best for the densities. Give him by the curve for the wires given by this pyramid. Carmine Kerf density given by the square of the distance which is that school prosperous crypt plus the square. And if your computers t squared plus one and the esus DX DT square Krusty, why did he score? Project his square And is it you can use a computer that should be square off too. And Titi and uh, before I start is problems already compute is too into grow. I mean these two. You can just do integration by part twice together. Conclusion I'm computed beforehand because I'm going to use it later. And I don't spend majority of the time in this problem computing the integration by parts. Anyway, let's see Soto Computer Mass. We simply have to hint a girl, Um, from zero to two pi times are hard wrote he squared plus one. Yes, his square root of TT All this math start swearing about Tootie Square plus square root of two. So swear it up to you as far easier to just put it outside zero to two pi. Thank you, Boris Reprocessed stall. This should be eight pi cubed over three cost you a pie. So that is a mess. How about the How about the center ofthe mess? So in the case of center ofthe mass, we have to Hey, Chris, from zero to two. Pioneer x Times room and access T Time's ro is t score plus one. And yes, is squaring up to D. C. Well, I put him here, miss a mess so as a and we have to divide by him. But do it at the end. Keep your pastilles and that gives us the form over for crossed the square over too. So this is sixteen pie for over four four pi for plus for pi square over too, which is to pi squared. And what is him? Him? Is this square root of two snowy, No hex like a sex party course This party, divided by square's too, can so out. By the way, I hate overs three pi cubed plus two pints so we can sew a pie to pie eight o ver sary Paice. Trust me. And, uh, now that's compute this. This should be a row times Why so in tomorrow from zero to pie drome times why is co sign t and he s squared of beauty. Here's where I need the form right The first page. Otherwise, I'll spend all the time to integration by Park Boys Co City So we have t square minus two site B plus to be cool So that's t square Holstein heap Arcosanti Part will be we'LL be negative Sign t the entire sorry positive sign t the anti derivative. However, this one If you point to pie, this will be cereal. This will be zero and this will you have. If you're trying to buy, the sign will be zero So this will just be for pie and co sign you will be one if a problem zero See Marie Science All the scientists would be zero and co signed term will be one stole There isn't anything that this Mr Interesting there Make sure if I get this correctly. If I buy two parties will be zero zero and for pie time Swan And if we province zero zero zero that's ours right there. There's no minus four But I got this wrong. Oh, probably zero Every time is zero beyond this room. We have a team from this should just be like this and issue these four square of two pi and And we have the formula for M. So my horrible just before squared off to pi over hate over three Pike. You plus two. And I think we can sew a pie And this computer Last term similarly, zero to two pi dro times scene He is a scientist and the yeses square with TT on t score side here. We also have a form. Now your side plus two minus his score. Coke side. Sorry. This should be from zero to two pipes. Get now. We have the square root of two times that. And, uh, the scientific part is nephew of co sign. So if we probably to pilot zero here and, uh, pulling into pie, will ya? Two minus four pi square times one minus one minus four point zero zero Uh, minus two with point zero zero plus two minus one year. Always should square it up too. Two minus four pi score minus one minus two plus one. So we also get Oh, yeah, minus four. Pi squared Times Square. The two. Sure. Yes, Oh, so the partial p this over over the mess square root of two. A pike. You over. You know, I think I forgot something here. You sure? Ready. Cancel a square root of to So sorry about that. Why? Partially this on the bar. Toby, This over three plus two pints. How can saw the square root of two? Only because you can sew one of the pine once you have a formula is just a detail computation.