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9. The integral expression |drcouldrepresent the arc length from * #ntow #2 for the function [email protected] B 2 c In( D h(} 2 In(+? )...

Question

9. The integral expression |drcouldrepresent the arc length from * #ntow #2 for the function [email protected] B 2 c In( D h(} 2 In(+? )

9. The integral expression | drcould represent the arc length from * #ntow #2 for the function [email protected] 4 B 2 c In( D h(} 2 In(+? )



Answers

Arc Length Name a function for which the integral below represents the arc length of the function on the
interval $[0,2]$ .

$\int_{0}^{2} \sqrt{1+(4 x)^{2}} d x$

In this question. Good record about Ackland function as they will be denoted by the integral from 8 to 30. The number under our prime new do you And now in this question, were given the factor functionality ego to the 40 about a half online on the D to the for the equal to one Here we see the first time we need to find a prompt e They lived in the fourth They will have equal to the far over two squared on the the and they reviewed them that the centipede Because you're one of a d they give them that you the energy to therefore we confined? No, on the Bronte it will go to the square it off now Here we have the tooth square Could you fall over the D plus one of the T Square thus far? Now we can simplify everything to bring them to the common denominator will be the square on the time we have the 40 plus one plus 40 square and we can ready Stand into the on the numerator We have the perfect square where we have the four to the plus one. Don't you off a t Square How we can write this one equal to the to the plus one on the D equal to the two plus one of a D. And from here we should be able to find the functionality you go to the in the glory. We have my ankle to one now to the D And then we have the two plus one of you now, do you? I'm Terry River them that you equal to that to you? Untidy, riveting. The one of a new coach. And I remembered you evaluated from one today. And we have if under, do you get to go to the to the plus online on the T on, then minus a woman. The one inside. We have a miner's till now, and that's gonna be the answer.

In this question. What we call about the Atlanta on the function Do you notify the SD every equal to the integral from any to the D The number on prom T the The on now in this question here were given the vector function rt ego to the the sorry doesn't use the new here to be more peace Precise you did you? And now in this question, were given the anti equity that t square to the square table three. Now we're given equal to zero here the first time we need to compute the arm Prompt e Yeah. Now then you live with them with the square, including the Tuesday from the second term we get including the front day from the third term getting culture the 33 square and therefore we can find a number on the arm from t It will equal to the square root off the fourty square plus 16 30 square plus the 94. We can write this down into the do you can bring outside and inside we have the farm. Last 16 will be 20 plus 90 square, and now we should be able to find a STD here equals integral, you know, equal to zero after the day. Then we have the square, This one You mean that you now square root on the 20 plus nine new square the U Now we're going to use substitution Where the Buddhist on being the I would call it as the vino ICO Children 20 plus night New square Devi Ego Thio it didn't you? Do you So do you echo to the d V over 18 You? I'm so when uh, u equals zero the V equal to the 20 when the new equal to t then the vehicle to the 20 plus nine the square Therefore, we can write this integral in terms of the vino from the 20 up to the 20 plus 90 square that we have the new square root on the V and d you echo to the d V off 18 you Now we can cancel the view on bring the 18 outside in the girl that we have one of 18 integral from the 20 to the 20 plus 90 square And we haven't a V about a half TV. Then we get equal to 1/18 Untidy reverted of this one equal to the Viva three over to David didn't treat him still from the 20 to the 20 blast 90 square. And then it was simply fine. We got equal to the one of the 27. This will be a STD equal with you. The 20 plus night the square 503 over two, minus the 20 about three hour Jew. And this will be the answer. Fond background function, Yeah.

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.

We're going to use the arc length formula to set up our integral. And then we are going to use our graphing calculators to actually find the value. So notice things are in terms of why we have X equals in terms of why. And then we have the Y. One to Y equals four. So for that I am thinking of my derivative of X in terms of why. So I'm going to write it as the integral of the square root of that derivative squared plus one. Okay so we're going to be integrating from 1 to 4 and let's go ahead and do our derivative on the side. The derivative of why? Because remember we're in terms of why? So we can just write why and then the derivative of Ln of Y is one over why? So we'll go ahead and we'll put that one plus one over Y squared in our square root sign and then plus one now um real quickly um when you have a integral, that's in terms of why you can still put it into your calculator in terms of X is once that inter girl is set, it doesn't matter what variable you use. So when you do integrate that you'll find that you get a value of 5.3-3-4


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