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Thls question has several parts that must be completcd sequendally. I( you skp part of thc quastion; YoU wlll not reccivc any polnts for the skipped part; and you w...

Question

Thls question has several parts that must be completcd sequendally. I( you skp part of thc quastion; YoU wlll not reccivc any polnts for the skipped part; and you wll not be able: t0 corne buck the sklppad partJutorbal ExerclsoAlnd the exact Iength or the cunyo 16(* 2) , 0 <*<7, Y >Stepcurve given by y = I(x), arc length glven by:Step 2We have y2 = 16( + 2)%,which can be re-wntten as follows.Stcp6(x +2calcP OpurulStep 4FuncloThe arc length can De found by the Integral;SymbolRulaloVocto

Thls question has several parts that must be completcd sequendally. I( you skp part of thc quastion; YoU wlll not reccivc any polnts for the skipped part; and you wll not be able: t0 corne buck the sklppad part Jutorbal Exerclso Alnd the exact Iength or the cunyo 16(* 2) , 0 <*<7, Y > Step curve given by y = I(x), arc length glven by: Step 2 We have y2 = 16( + 2)%, which can be re-wntten as follows. Stcp 6(x +2 calcP Opurul Step 4 Funclo The arc length can De found by the Integral; Symbol Rulalo Voctore Subrit Shid (Vo Cannn Nora



Answers

Find the arc length of the following curves by integrating with respect to $y .$ $$x=2 e^{\sqrt{2} y}+\frac{1}{16} e^{-\sqrt{2} y}, \text { for } 0 \leq y \leq \frac{\ln 2}{\sqrt{2}}$$

Units. We're selling you right here, so we have to find a link that occur for why is equal to eat. The X on access between zero and two included, so are derivative because this e to the X So using our darkling formula, we just plug it in since from 0 to 2 square root of one plus eat the X square T X in the skin says negative square root of two plus square root of one plus each. The four plus the hyperbolic function of 10 in verse square root to minus the hyperbolic function of 10 in verse square root of one plus need to the fore power.

Okay, This problem, what we want to do is express the arc length as an integral for y equals X to the forth between X equals to annex eagle six. And if you went back on pages for 67 to 4 68 the formula hair shone and red was derived for finding the ark length. So what we need to do is express our circumstances in this form. So first of all, let's kind of get a visual of what it is we're looking at. We have ah, version of X to the fourth right, which looks something like this, perhaps. And what we're looking at is from X equals 2 to 6. We're interested in the arc length of that curve. So what we're trying to identify is the length of this of this curve here this ark late. So what we need to identify in order to fit the formula hairs we need first find our limits of integration and that's fairly easy. They gave those to us here as two and six. So we're interested in the arc length from 2 to 6. And what we have other other piece of this puzzle that we need is the derivative of the function that we're looking at. So what we want to do is we want to take our function, which is f of X equals X to the fourth and find the derivative of that function. Derivative of that function, of course, is for X to the third. And then we're going to need to square that within the formula. So if we take f prime of X or the derivative of X squared were basically taking four X to the third square, that means we're gonna have to square the fort and the X to the third. Of course, that means the same thing is, if we wrote for X to the third time's for X to the third. Therefore, we've got 16 x to the sixth. Is this f prime value here within the interval? So to complete the integral, then we basically simply need to fill in the pieces. So we're gonna say the arc length that we're interested in is the integral from A to B A. In this case, of course, is too, and be a six. So we're going to say the arc length is from 2 to 6 are limits of integration there. And then we're going to say the function. Val, you're there. Were the the thing inside here we're going to take is one minus f prime of X squared, which we decided was 16 x to the sixth. And of course, I've left this off inadvertently. But we should have RDX there, so this Integral doesn't want us to evaluate it. But that in a girl will give us the arc length for y equals X to the fourth from X equals 2 to 6.

Okay we are going to use our length formula to set up this integral and then we are going to use our graphing calculator to find its value. So first notice that we are X equals in terms of why? So when I look at my formula always think of like a X squared component plus a y squared component. So when I'm in terms of why I think of doing my derivative in terms of y squared plus the one squared. But obviously you can still do one plus because you know you can add either direction. Okay so um with that we are going to need the derivative of Rx and we are integrating in terms of why? So we're going from 0-1. So the derivative in terms of why um is going to be ex prime equals the half comes down. I'm going to rewrite my four minus y squared. Now I go down to power to a negative one half and then I multiply by the derivative which is a negative two Y. So notice my two is cancelled out. So I'm just going to clean this up and I am also going to square it because that why can become a Y squared and then that four minus Y. Now can just be to the negative one. Power meaning I can just write it on the bottom. Okay, now, even though we're in terms of why once you've set up an integral it doesn't matter what your variable is. So you can put that into your calculator with exes in the places of the Y and still integrated to find the correct answer. So when we do integrate it, we find that the value is 1.04719 it.

So we're told that wise go exit 1/4 over four plus worn over a X squared and we are asked to find our planes. So to start off with, we need to find our derivative. So you know, there are derivative. She goes, Why prime using? I don't see So we have 1/4 exit a force which will just get us execute and then minus 1/4 x cute. All right, so now that we have that next thing we can dio I was weak in front are quite so now he's able to square it one plus x cubed minus 1/4 x cubed squared d x So next thing we can do is we can apply you substitution. Where is us equal to x cube? Do you to go to three x squared D X and V X is equal to when I was three x squared you and so that will get us on. This show would integral from 1 to 2. So get us into girl from 1 to 8. Uh, for years, Word plus one over 12. You to defy search. Thank you. We can factor out 1/12 to get us. Did you go from 1 to 8? Four. Use word plus one over you. 2 5/3 do you? And so then? No. For you. Evaluate. Both of, um, we can actually separate them. So we have 1/12. It's integral from 1 to 8. A four year to 1/3. Do you plus inch girl from 1 to 8 of one over you to fasters. Do you? So it's cried out for a little bit more room. Okay, so now the next thing from do is we 10. Okay, so we have one tall. We can use a reverse part roll. So we have not see four four divided by Is it for over three focuses three u two for over three once, eight. Plus I seen they give 5/3 and gets us. So we have naked. If I start one over you too negative. 5/3 who boost it. We get negative to first get minus 3/2 times you to negative 2/3 from 1 to 8, which will get us a ceases to reacts to for three years to the 4/3. Oh, that's six. I'll get us Workforce. Holmes, 48 minus three. Then we have negative 3/2 x or three over to you to negative 2/3 which will gas, etc. Plus you're three ace minus nine. Victory over to She has won over 12 times. 48 close. My ISS, which will get us 123 over 32 has our answer.


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