5

Find the exact length of the curve l = In(1 - x2); 0 <* < 11Submit Answer...

Question

Find the exact length of the curve l = In(1 - x2); 0 <* < 11Submit Answer

Find the exact length of the curve l = In(1 - x2); 0 <* < 1 1 Submit Answer



Answers

Find the exact length of the curve.

$ y = 1 + 6x^{\frac{3}{2}} $ , $ 0 \le x \le 1 $

Hey, it's Claire. So in your radio. So we're gonna find the length of the curve for why is one minus e to negative X excess between zero and two. So the derivative we get is e to the negative X and our length Argh! Link formula from 0 to 2 square root of one plus e to the negative two x d x We're gonna substitute you square for one plus e to the negative to X So when we substitute to you, do you equals negative to e to the negative two x t x d X equals negative. You do you over you square minus one. So this changes the limits of integration and they become from negative to square root of two. Excuse me to one plus e to the negative four you times negative. You do you Over. You square minus one. We go here, this becomes equal to square it to square root of one. Pull us me to the negative for use square, do you over one minus you square which is equal to square of two square root of one plus e to the negative four you square minus one plus one over one minus you square. And this splits into two. So it splits into you. Square minus one over one minus. You square from negative too. To one plus me to the negative for do you plus read too. One plus eat the negative for do you over one minus You square We simplify this. This becomes equal to negative. You too. One. Pull us Eat the negative for do you plus route to one plus b to the negative for do you over one minus You square will continue. Here comes negative rich too. One plus e to the negative for D u plus 1/2 rich too One plus eats the negative for one minus you plus one plus you over one minus you times one plus you Do you When we simplify this this is equal to negative route too square root of one plus eat the negative for do you less 1/2 square it of two square of one plus eat the negative for becomes one over When mine issue plus one over one plus you do you And this becomes you When we ent I drive it This becomes l on negative l end of one minus you. Class Ellen of one plus you. And when we find this, this is equal to square root of two minus square root of one plus eat negative for plus 1/2 L n of one plus square root of one. Pull us eats and negative four over one minus square root of one plus Eat the negative for minus 1/2 well and one plus square of two over one minus square root of two. This is equivalent to around 2.22 one for two.

It's Clara. So when you read here, So here, we're gonna find that exactly. We're gonna first start with the derivative. And we got negative two acts over one minus x square. We're gonna use this on plug it into our our Klink equation Square root of one plus negative to X over one minus x square square. When we factor this out we got four x square all over one minus two X square less x to the fourth when we're gonna make one bye. Using a common denominator two X square, it was four x because X to the fore. Excuse me one minus two X square plus X to the fourth. And when we add and simplify, this part becomes from zero to wouldn't have one plus x square over one minus x squared DX. Because we're taking this square root off the integral. We're gonna defy the top and bottom. Using long division in this equals from 0 to 1/2 negative one plus two over one minus x squared d x and we're going to use partial fractions. So a over one plus tax must be over one minus X or finding A and B and we get a to be one be to be one. So it becomes negative X plus from 0 to 1/2 one over one plus x plus one over one minus x the ex. When we just integrate this, we got negative one have plus Helton of three.

Hey, it's clear. Someone new right here. So we have lying is equal to X cube over three plus one over four X. We're first going to find the derivative. So do you. Why, over DX We got X squared minus one over four X square. Afterwards, we're gonna square. So we get. Do you want? Over. Deac Square is equal to X square minus one over four square square. Um, excuse us X to the fourth minus one, huh? Plus won over 16. Next to the fourth. Now, when we plug it into our arc length equation, yeah, l is equal to from 1 to 2 square root of one plus next to the fourth minus one have plus one over 16 x to the fourth DDX. This becomes X to the fourth plus one Have plus one over 16 x to the fourth. This could be factored into X square plus one over four. Next square square. So when you put it into our equation comes from 1 to 2 square root of X square, plus one over four x square square. The this equals 59 over 24. Yeah,

First will want to solve for y so immediately we can see that we'll need to divide by 36. Getting why squared equals one over 36 times X squared minus four. You then we can square root both sides to get why equals 16 times X squared minus for to the three seconds now Because we are given that why is greater than or equal to zero we will only need the positive of the square roots. We won't have to worry about the negative one. You can go to our next page here and well, you will need to take the derivative of why so we can see that 16 times. Three over two times X squared minus four to the 1/2 now and then times the derivative of X squared minus four, which is two X Now the twos cancel. In this case, three over six is going to get this 1/2. So why prime equals 1/2 Times X? Because we have to remember our X over here. Times X squared minus four to the 1/2. So going over to our next page, we have our derivative. Now we can plug this in to our equation. Um, we were given the interval two three so we can plug in to for a on three or b radical one plus 1/2 x times X squared minus four to the 1/2 squared D X. Now let's go back for a second. Um, we will square this. So this is 1/4 x squared times X squared, minus four and then squaring. This eliminates the, um, 1/2 power here. So if we go over here, we end up getting 1/4 extra the fourth minus four x squared. Now we'll go to our next page, and we can factor out 1/4 and then we can move it outside and reorder terms the bit. And now we have X the fourth minus X squared, plus her minus four x squared plus four d x. And we can see that this can be also written as a radical X squared minus two squared D X. And now, um oh, I've already written that here. Oh, no. Okay, pardon me. So in this part will simplify um, rather integrate. So x squared will be x cubed. 1/3 minus two x now um, still with our boundaries. Um, 223 Um, And then we'll come to the next page where I've already written this out for us. So this comes down to 1/2 9 minus six, minus eight over three, minus four. And then, uh, simplifying about more. We have nine minus six equals three minus eight over three. Um, you have plus four. Distributing this negative equals 1/2 seven minus eight over three. Um, we'll continue to simplify. So this way we can write 21 over three for seven blindness, So this will give us 13 over three times 1/2 which equals 13 over six, which is our final answer.


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