find-the-exact-length-of-the-curve-yln-left1-x2right-quad-0-leqslant-x-leqslant-frac12

Question

Find the exact length of the curve.
$$y=\ln \left(1-x^{2}\right), \quad 0 \leqslant x \leqslant \frac{1}{2}$$

Rakesh Kumar Sharma · Accepted Answer

What? So we have to find out the government for the call, which is you know, my Why did you go? So let off. One minus X is where exes from zero half. Okay, so we have to find four back explosives are differentiate the stop. Get it first. So why does reason ticket the position off Henry V that Are you going todo So there is one minus X expects. And the sensation off this part? No go. You&#x27;re going to go. Use equals indignation Zeroes off room door one bliss full excess square before I need May one minus. Exists where? You know, starting in. Do the office. Okay, now that&#x27;s for this. If you fight missing, so does your mom. You know, I didn t fail our biggest even open. This one doesn&#x27;t want to get any beginning one bless, except for my constituents, is where oh, bless for excess. Where and then they missed the nominal keeping says this one minus x elsewhere. And it&#x27;s quite off that window in the office. Okay, listen, if I exert dreams, you may do indication so sincerely you can see this miner strikes is where it gets along with this elite unit or dignity on press accessible for us to excess way. And there is nothing but they swear off one bless excess. Where? What is right over there? Okay. And the report on the hook over there. So do you ideally want brightness? It is where hazardous you do? Yes. Okay, Cincy, it will become indications you I think my this comment from that isn&#x27;t so. You have this squarely you can see over this is where it&#x27;s quite literally. This is where you can seeing out with. I hope so. What will be impotency? BF one Bless. It&#x27;s a squared B. We didn&#x27;t wait one minus X elsewhere. You do the affects. Okay, Lets so many partner. Let&#x27;s all the sexy girls give up. Okay, So they do. You have a very d&#x27;innocenzio zero cough. I think this negative comin up off this Still that become excessive? What? I am doing this. Do you? It&#x27;s just where this one I suspect anyone. We should be frank further And in that I do. I am making excess for justifying. Nothing else. Now you said you fight so minus indications here. Do you think this excess credit This one&#x27;s for you If you take this excess fire with this minus what&#x27;s in the military? Become extra square minus one, divided by excess. Well, right, So so that you become one. And you left with a little our way. It&#x27;s a square minus one. Okay, You know, you fix further for integration. Off will be selectively. They didn&#x27;t know so Indication off one recently X bless who will visit this? Okay, an indication off one of all our X ascribed my Muslim will be. Oh, one way in this case is people so that you want upon you in. Okay, This one and then all X minus one. You were lead away extra small and ward off that okay. And report the liquids has lots of lower disease of communities off. No. So if I need to put the parliament so limit will be minus O that this too And this to really because, you know, alleviating. So that really bless If you put half here so it will be half minus one off minus one will be minus half and delivered a dream come true for days in more solder minus milord. Having an effect certificate unity on my do. So when my when my do you cancel out. So we be getting yet and, you know, minus we got all over that issue. Okay, So that this lower limit is zero for the year. So your very being zero. And if you put zero So all zero minus going there. And residents also, if your phone doesn&#x27;t define a few years ago Property of lentils. You getting this team? Newman? It is beginning. Bye. Miss off. We miss you and father a few $100 of your beauty. Let&#x27;s see my off and their designs.

Robert Daugherty · Answer

problem number 73. So we&#x27;re dealing with problems that requires to be able to do partial fraction decomposition. Here they ask us to find the length of a curve along the length along the curve, like with the natural log of one money&#x27;s X squared for X values between zero and 1/2. So we&#x27;re guided by our formula for art went over An interval is going to be the square root of one plus the derivative of that function squared. So what we need to do in this case, we&#x27;re looking at a function f of X is equal to the natural log of one minus x square. So what is the derivative of that function? The derivative is going to be 1/1 minus X squared and then using the chain roll, the derivative of one minus X squared is what, minus two X And so this case you&#x27;ve got minus two X over one minus X squared. And if you just sort of like to not deal with that minus sign, I could just write. This is two X over X squared minus one. So that is by derivative. Now I just need to plug that into the formula that we have. So I&#x27;m gonna need to figure out then what is the square root of one plus f prime of X square? So I need to work on one plus what we say. The derivative, the derivative was two x over X squared, minus one squared. So now let&#x27;s just get a common denominator here. The common denominator is X squared, minus one squared. So you&#x27;re going tohave X squared, minus one squared plus two x squared. So this is the square root of X to the fourth minus two X squared plus one plus four X squared over X squared minus one squared, which is the square root of that&#x27;s X to the fourth plus two X squared plus one over X squared minus one squared, which is X square plus one squared over the square root of X squared minus one squared. So what I&#x27;ve got at this point, I&#x27;ve got that the art length the art link that I&#x27;m looking for is going to be the integral from zero toe 1/2 of the square root of X square plus one squared over X squared minus one squared DX. So this is the integral from 0 to 1/2 absolute value X squared plus one squared minus one DX. Now, on the interval from 0 to 1/2 you know that the numerator is going to be a positive numbers. You got a positive number plus a positive number. So this turns into the integral from zero to 1/2. I could just get rid of the absolute values there. Now, if you look at the denominator on the integral from 0 to 1/2 that number is going to start it. When I plug into zero, it&#x27;s a negative one. When I plug in a 1/2 is a 1/4 minus one. That is always a negative number to take the absolute value of a negative number. You just take the opposite of that number. So X squared minus one DX. So this is, um, minus the integral from 0 to 1/2 of x square, plus one of rest squared minus Wendy X. So that is what we&#x27;re tasked with. Now it&#x27;s finding the interval from zero minus from 0 to 1/2 of X squared, plus one of Rex, where mine is one DX. Now I&#x27;m gonna leave you to do this. If you&#x27;ve gotten this far in these exercises, you should be quite comfortable with doing partial fraction decomposition. So X squared plus one over X squared minus one. It&#x27;s simply going to be one minus one over X plus one plus one over X minus point. That is the partial fraction decomposition, and you can go through the process to end up with that. So what that tells us is I need the integral from 0 to 1/2 off, one minus one over X plus one was one of Rex minus one DX, and now all of this easily integrates. So this is going to be minus when you innovate one who just going to get what X and then minus natural log X plus one plus natural log X minus one and then all this. It gets evaluated from zero to 1/2. So let&#x27;s go do that on the next page. So I&#x27;ve got minus X minus natural log of X plus one plus natural logger X minus one and this evaluated from 0 to 1/2 so minus X, Um, and that you could just write this plus, um, natural log X minus one over X plus one from 0 to 1/2 and that this is now. We just turn this into arithmetic problem to get to our final answer for the argument. So let&#x27;s do the substitution. Said this is going to be minus some quantity minus minus some quantity. So let&#x27;s substitute a 1/2 when you substitute a 1/2 and x, you get 1/2 and this is plus the natural log of so 1/2 and over three halves. Typically, the minus sign is going to get caught in the absolute value. When you substitute a zero in, you&#x27;re going to get a zero and then you&#x27;re going to get plus natural log at survey of minus one over one. So you know that this term is zero. This term is zero. And so what you end up with is minus 1/2 and what you end up with in this case right here, you&#x27;ve got 1/2 over three halves, so this is just 1/3. So this is plus, So is the natural log of 1/3 which is minus. You got 1/2 you got plus natural law go one minus natural log of three natural Aga one is simply zero said This final answer is minus 1/2 plus the natural log of three. So it&#x27;s quite a bit of ways, but that is the art lent. So for this curve from 0 to 1/2 the art length is natural. Log of three minus one halfs again. What we did was usar art length formula. The art lent formula lead us along a path that where we needed to know what a partial fraction decomposition is. Once we get the partial fraction decomposition, the integration becomes straightforward, and they just substitute the value 0 1/2 to get art length of that curve.

Rakesh Kumar Sharma · Answer

Well, there&#x27;s she looks on the question. You find out. Yeah. And the carbs. June is Why is he postal one minus just one minus six And X is different from your room. You know, what we&#x27;re doing is leaving me if inserting the force semester contribute toward rebuilding, even getting worried. This is the position of the millions ago. And deposition, Lex or minus here. The physician off is tomorrow. March 16. And now the position of where sexually? Me minus one. So hopeless for your for your function right now. Let&#x27;s hear from you. Wants to go for always Medication? No. What? Unless first did your This quack really has to 1 26 for this. Where this was meat is too far minus to explain the Oh, this Do you before them What we&#x27;re doing this is You mean? I know that this this is really the underfunded. So, like one plus either Sabata minus Pruetts. It&#x27;s supposed to be sweat your defensive Gumbel&#x27;s toward the traditional media and the physician off a list of all my students to be. It&#x27;s about minus place in the transition off minus two X would be What is this? OK, but the physician off my sister Be acceptable. Moving what? This is also for me and we can find out the really off leadership are my districts beatings to Baltimore? Yes, two weeks. Maybe it was so okay. Lets for these values back in this Indian. Right? So I recent shooting? Yeah, I&#x27;m keeping it as of this. So indignation. Oh, the very off this thing when my skittish one dynasties minus would be east way. So I was literally containing the splashes and and from the extreme of the extreme right there with me. Okay. Big day. Be more daylight. He just for my stores. And he asked for my instructions. You see, peace you slam this minus So for those before, so I would be You can be my disciple aspect to this maybe century on being big. He scratched this with me. Use flair. Wait, we swear. You know what I do is I know Subtract one of the numerous and I am Everyone is essentially your weekends or off with friends. You go. You just do. Why doesn&#x27;t define so likely beginning this Misty&#x27;s four minus one day I read these for Mike Scientifically become once like Steve this name or did you let Inglis? Oh, he&#x27;s from I was wondering, really? Someone wants one. And we have only one use for arthritis for you know? So let&#x27;s indicated so the indication off this my guess. Okay? In addition, one really d And in traditional support this for my one of all who will be easy and so that I know he just wanted the rest. One more. Let us know. Even if you&#x27;re gonna be zones. You see these? What is this for? Beginning the very off ps don&#x27;t want my lesson is what, plus years, four months to get. And maybe something that we live in an apartment living as so distribution, shooting How much? Everything. We had an empty so that this What was he Just one morning This prolix Don&#x27;t go there. That this tree of your team This one by one Morneau You know, with this us about minus flicks by in this life this way. Do you mind everybody? What was he just about minus two ways, This one? Well, let&#x27;s get that waiver in its A minus. What limits stew for getting No, no, no. One plus years about minus full. Yes, What I do? Well, it off Mobile. One of those he justified. Minus four minus one. Be right away, Mugo. One plus English for minus full. This one. Okay, so there is a parliament and the Lord would remind this like you put Zito set up X 40 years less When my do anything like this, like I do for you. So they go school minus one. Who s one more than that? Yeah. Okay, so there is no fire. So just backing years off. This is called immediately. Give

Clarissa Noh · Answer

it&#x27;s Clara. So when you read here, So here, we&#x27;re gonna find that exactly. We&#x27;re gonna first start with the derivative. And we got negative two acts over one minus x square. We&#x27;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&#x27;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&#x27;t have one plus x square over one minus x squared DX. Because we&#x27;re taking this square root off the integral. We&#x27;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&#x27;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.

Rakesh Kumar Sharma · Answer

Well, so? So you should find leg look okay. And the ocean waas essentially is he goes through Nexus. Where by four wearing this for the next meeting. That excess strong when you Okay, so for the first differentiated this stuff because you&#x27;re supposed todo me who actually went anyway? Four minus the petition off Next. What works? So you re getting too. So the first year we are going to have this one way to do, you know, explain this one, but it so that it is often stated you can get the left, so OK, indignation oneto more dollars. What&#x27;s first? Earlier? This one And they disposed of your just that Really SimCity one by four Explain this one by X What is where we will be others before? That&#x27;s a defiant towards your genius Indignation where you grew Just if you find a way, we&#x27;re doing this one less nexus whereby four I&#x27;m ready lanes When before is it? Yes. One bite four Texas Where? Ok minus to exit Doing wakes xx guys love to read it to by force of that one body. Thanks for this. Great. So we&#x27;re getting an indication we&#x27;ll go. You see every building way having so what might this have become comes if they don&#x27;t become option? We can convert or reconvert This impacting is one by foreign boom Next less one by excellence where you don&#x27;t have to leave it Ok, Please see my don&#x27;t be my father made retake that are take out some way to get someone You take over Michael Tree and this square weekends alone with this and dems will be sitting a big X Westerlund bakes You know your ex looking Well, this is your You&#x27;re getting it when I do indignation off Extreme beauty Sentry excess Where? Bergen&#x27;s last indication off You have to lower every day for just two days. It&#x27;s your father. So what? We&#x27;re giving you copper Liberty Freeport? This I do minus 11 people is one. So one way to less. Okay. Your father doesn&#x27;t afraid to boot. Minus off would be off to it. Do you sincerely? One by two we don&#x27;t even do. Plus a little. Do you food? Bliss? I don&#x27;t like that

Rakesh Kumar Sharma · Answer

So we have to find out that Linda, look off. Mexico, Antarctica. And the card given to us is actually why? Because too excess Where? By four minus. And the next day four x belonging to one who Okay, ah, selects defense rejects a little beginning with my desk will be closely together First they&#x27;re dreaming weeks my four minus when a ball works This two and four have been slow to you creating this two years You can take one way to common. So we&#x27;re getting actually x when we don&#x27;t say and you will be getting will explain this one. You know, Let&#x27;s so far for land. No go Mrs was so integration Want to do going one place first aired you won&#x27;t spare. Supposed to reduce this? This is a message that really even Michael&#x27;s car. So it will be won by four into x minus. When my exclusive with you don&#x27;t be ethics. What makes you think if you should be afraid for you Dingus indignation that really close to when you do intuition One place who sweat out this extreme before this wonderful for excess wet and my listen, America, if you open it and I think that&#x27;s what And go the f it&#x27;s OK. Well, listen, if I actually getting one to do who go, this will be actually exists whereby four This would be one way for excess wear and this one in minus half we could together in my staff. So your vehicles and then in fact, over become one before you know, extra us When my explosives question that that that don&#x27;t become like that An indication on there and even give you See if you take a poem by four I&#x27;ll turn up simply go. So we&#x27;ve regarding this indignation want to do We&#x27;ve been taking with middle turn on the deal and this experts on May 1 square We can tell you this a little too in this impunity Next this one breaks into be a fix Next for those of you fight so integration when you do What were your hopes? A indication? All fixed Maybe so X plus one by X bx The indication off This will be indication off X Revis in Chile. It&#x27;s just fine. We do an indication off when my excuse me well enough. And we have to do What did you access it. No limit is one apart from a distance. Okay. The reveal beginning in this one by two export. The values are delivered. This to support two years to push. Whereby so That revision chili who? Yes, but acceptable to do your civic beauty and then do minus naked exit. You also want to be bigger when I do and put actually goes to one side will become CEO, so that will be blessed. Okay, which should be afraid further. So this two and minus one will become to minus half the country by force. So that maybe one body to seaway go plus Ellen. So if if you see if I completely so three by four this and you do your right toe so that would be on

Problem

Find the exact length of the curve. $$y=1-e^{-x}…

Problem 17 Medium Difficulty

Find the exact length of the curve.
$$y=\ln \left(1-x^{2}\right), \quad 0 \leqslant x \leqslant \frac{1}{2}$$

Answer

$-\frac{1}{2}+\ln 3$

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$-\frac{1}{2}+\ln 3$

Essential Calculus Early Transcendentals

Grace H.

Catherine R.

Anna Marie V.

Kayleah T.

Integration Review - Intro

Definite vs Indefinite

James StewartEssential Calculus Early Transcendentals

Grace H.

Catherine R.

Anna Marie V.

Kayleah T.

James Stewart

Essential Calculus Early Transcendentals