find-the-exact-length-of-the-curve-y1-e-x-quad-0-leqslant-x-leqslant-2

Question

Find the exact length of the curve.
$$y=1-e^{-x}, \quad 0 \leqslant x \leqslant 2$$

Rakesh Kumar Sharma · Accepted 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

Enkhzaya Enkhtaivan · Answer

four from 8.4 and we&#x27;re giving function. That&#x27;s one minus eater nave axe on domain. And the question is no length clank. Oh right or this doing this work? So it&#x27;s to a brief sketch, sirrah. Why thanks? So at zero, our function has value 01 Our function, its valley. One lies he urinated one rich. So we&#x27;re girls are functionally look like something like this. You can take intuitive dysfunction and realize that it&#x27;s positive. So it&#x27;s increasing in the formula Darkling form What is this? What is called a length l So this is the formula in itself is not simple words Integration off skirt of one plus like crime X squared DX So in our case, this has become This will be killed one here to warn you screwed off one cross. Why Primex? So our function is one minus steed through negative X, so live prime here it&#x27;s going to be negative. So zero minus minus the two native X or simply so. This is need thio nerve to x d. X here and so it&#x27;s right this way like this since we have either the native to x right at us a 22 x plus one Mind tonight&#x27;s DX or simply it&#x27;s become sirah to barn Screw off Peter two X plus, plus one to exterior. So now this calls for the first substitution. Where we will do Entry X is equal to oh team. So then so since exes go from 0 to 1, tea will go from Eat Reserve Ridges one and the two first powers That&#x27;s Just and d T I was reading E X years this It&#x27;s basically tee times DX here. So after this transformation, we burned to have into Let&#x27;s go from That&#x27;s never want to be screwed off us T Square derided by this t that&#x27;s already here. And then one more ti will come down from the differential, So it&#x27;s kind of a T squared here. Now again, this calls for this calls for Chris Constitution. So whenever we have one plus t square, we our first candidate is tease tangent. Churched. It&#x27;s tin Geant. Oh oh, it&#x27;s it&#x27;s called, uh, fine. Yeah, so now says he&#x27;s going from one to he engines slow bound Is Thies attention off. What anguish One will the estrus pilot for so five fires going from power for two. So now there&#x27;s E. We have e on the upper bounds. So for that we have any eyes value except for writing tangent. Universe off E. Now this is This is just the angle. So it&#x27;s called this angle um, my Alfa for piece of invitation for now and again. So it&#x27;s finally into computer differential. So did he. It&#x27;s going is equal to doing their attention to riches. Second square If I tio fine, so our internal is going to become pi over four Alfa and now one scratch from a dusty square by our choice is second of five, but their front differential DT We have two more parts coming, so it&#x27;s right this second Cuba fi defy on the bottom. We have t squared, which is Tangent Square here. So now we never again. Whenever we have a second contingent. Not not a literary but divided together. It&#x27;s awfully good idea to write this in terms. Of course science. I so stew that that sorta substitution and simply that&#x27;s simply some magic algebra. Here&#x27;s a second is when I would go sign. Tension is a sound course signs. After we put that into here, I&#x27;m going to have. Okay, fine. Divided by Science square. Fine and co sign Fly here. Now hear this calls for another truth substitution. So you want to substitute that Either sign of fire, Karsay if I sort of variable. But since we have sign squared here, it makes sense to say it&#x27;s a sign off. I is equal to some number. Are you? So then we&#x27;re gonna have to deal with sickle to car. Sign off. Why? Define right And then now you the balance for you. Well, the long bond we have sign of final for witches were not squared, too sort of going. And whenever squirted too good from you. Grew from sign of Alfa and Alfa Remember, al phase this specific value tagine in Versace. So it&#x27;s for a time being Denote this a simply sign off, Alfa because this is again just costume number And now we transfer him are in through into telegram from Hanover Square too. Sign off, Alf, I now we have de fine. That&#x27;s we put fine defy from here it&#x27;s going to be a It&#x27;s going to radio over course I If you so it&#x27;s d&#x27;you hear and the co signed a fire will comes Come down to the bottom So that makes it I&#x27;m signed squared If I co signed square five there. It&#x27;s a sign If eyes you were going to have you squaring unless you&#x27;re square here Now finally, we have a rational function that we can Oh, do you? For which we can do a partial fraction decomposition So So it&#x27;s actually focus on the interim now. And this is our This is the length. This is the length l we&#x27;re trying to find here. So is focus on the year Uh, they&#x27;re in function here now. We can proceed the usual way from partial fraction the composition. But if we observe a little little we should do a little observation and we can write one ass one last you squirt plus u squared. If he noticed this today, we can easily right dysfunctions of some off one Elias. Your square cards here squared and we know Danta jury took one of my issues. Square diesel. So now we focus on the second term and so it&#x27;s going to be won over. You squared now. One wise your square factors easily won likes you one. Plus you And for this term we do this again. We noticed that one physical too 1/2 off, Plus you when I have both one fine issue. So if you noticed this, then this work in the end Turn out to be whenever you squared Plus 1/2 off underway by one minus you cross one the right one Plus you Now all this three functions we note and to do with it. So now we can finally certain start start to violate our into so dismissed and l is equal to Oh and touch your ear off while different threw off this sign off, Alfred, whenever you&#x27;re squared plus one house you to you And that continues on the third page. So now for the first time I won over you squared that is the director of there simply anyone after you and the second and third term keep in line. Both of them would give natural work. So this whole and we combined them Go have one. Plus you write one wise here and now the balance We&#x27;ll never square too. And we have sign, you know, for here now This is where we should evaluate What&#x27;s the actual values This guy signed here. So sign of Alfa and we denoted Alfa is tangent inverse of e So this is sign off Tangent universe off a number This no through many trigger an insidious involving the relationship between trip functions and their interests functions And this will the kung he derided by screwed off is squared plus one So basically the formal I&#x27;m using here is sign off Tangent in verse off X There&#x27;s X absolute value divided by excreted by script for swat. Now this is a This is the simple form like that You could just look out from here text. So now we know what this number is And so l our internal becomes what its play in the 1st 1st value here. You know, I have negative one over sign of Alfa, which is this We&#x27;re here. So it&#x27;s going to be squared off for us to do right by me. And now one house natural log off one plus he divided by war brains Hey and subtracted. Now we have not Lower bound is simply simply one over square of Tuesday. This is negative squares to us, your health natural or off one plus one over square, too. My mind. So this looks like very messy number that can&#x27;t be simplify for. But there is a way to simplify this. A bit simplifies into a bit better. Look can finally. So for that, we had to basically work on what is How can we write this 1/2 times national art form? Plus, you thrive on lines, you. So it&#x27;s it&#x27;s working that here. So in general, when you have one house off natural, aren&#x27;t you? What&#x27;s more, my issue Miller 1/2 we can observe 1/2 into the inside that radius this and then we can do the water playing Bray conjuring trick here so we can write. This is now one plus. You square it off one lines. You square here, Andi. So using this. So now to simplify some some space. That&#x27;s call this mess, eh? And it&#x27;s called this smaller mess. That&#x27;s B. Then we could see that a scream to be equal to when vehicle to natural log. Now you can check the algebra with aid. Terrorist ends up becoming the actual ark of squirt of e squared, I sworn crusty after all, a simplification and B or recount natural work off spirit of to Class One. And you know what&#x27;s also interesting is that this has a underway to write. It&#x27;s actually inverse. You&#x27;re hyperbolic sign at the Valu E, and this is a legal too universe hyperbolic sign that simple act one worthy general close formula for his university board sign and Value X is no that sure took off X squared plus one plus X. So with all this in mind, we can finally write our allies Well, there&#x27;s a phone we can write easier terms first. So and I have squared off to Linus Squirt of P Squared course. One divided by and then we&#x27;re gonna have a C minus is plus a and minus B. So fitness class Denver Centre for science writers Evers had to go exciting off run. So there it is, the length off the curve that, scared by a rather simple function, turned out to be something that that&#x27;s that&#x27;s not the simplest looking number here.

Oswaldo Jiménez · Answer

in this problem What we want to calculate the length of care of given by y equals one minus six to the negative X for expectancy round one. We know that for, uh, curve given by the equation y equals ever vex Where if he said, differential function on on Interval A B, we know that the lance of the cure ofhis given by the internal from a to B off the square root of one plus two derivative off F square differential of X So in this case, we have effects even to want miners e to the negative X and so the derivative off F sequel to into the native X and then the length of the cure given by y equals one minus eternity of axis between Syrian wants seem to grow from 0 to 1 of the square root of one, plus he to the negative X square differential of X, which is the same us they integral from Syrian to one of the square root off one plus e to the negative two times six differential bakes. So this is the interim. We got to find the first. We&#x27;re going to find the ah indefinite integral that his intra loved the square root of one plus e to the negative to 10 sexy French Olympics. And for that we air going to make a first and change your verbal given by u equals to eat the native eggs. With that, we know that differential of you is negative e to the negative X differential of X that is equal to negative, you differential of X and so differential of X is able to negative differential of you do but about you. And so we get that that the integral off the square root of one plus e to the negative to tens eggs differential of X you see equal to the integral off the square root of one. Plus this expression here is equal to e to the negative eggs square. So is you square and differential of axis negative differential of you d better about you and this turns into negative being to really love you times This sorry turns into the your love is square root off you my, uh, glossy square divided by you you for rental of you And so we&#x27;re going to calculate now this integral you seem to gonna Magics institution, and for that we&#x27;re going to define implicitly available theater through the equation. You equals U equals to change into view dimensions of the anus or for an angle theater in the interval. Negative by have by half. With this conditional theatre, we can invert these change of our over here anti differential of you sequel to the second square of the differential thinner. And so we have that the integral of the square root of one plus e squared, divided by you Differential. A View, which is this internal here is equal to the Inter Loved, the square root of one last tangent square of theater, divided by tangents of theater times differential of you that he&#x27;s seeking square of theory. Differential of Theater, The sequel to the internal loved the absolute value of second of theater time seeking square of the differential theater to by the by the intelligent of theater in fourth era and this interval negative I have I have we know that the co sign of feel It&#x27;s positive and so is the second of theater, so this is single, too. The interval off the seeking to the third of theater differential of theater divided by the Tangent Theater, and we know that these sequel to Want Divided by co sign to the third of theater. This corresponds to second to the thorough theatre times Go Sign of Feta, divided by sine theta, which is wanted by the way, 10 gentle filler Differential theater. And now this is equal to the internal of differential of theater divided by Go Sign Square of theater time saying of theater. If we multiply both numerator and denominator in this Inderal by a sign of theater, this becomes the integral of sine of theater. Inferential of theater divided by Go Sign Square of the time sign square of theater, and this is equal to the interval of Sign of Theater differential. Theater Anybody by Go Sign Square of Theater times one minus Go Science Square of Theta You know we&#x27;re going to make it Third change of vulnerable here, so now leads to find the viral. We&#x27;re going to call it Z equals two saa go sign of data and then we have the differential Aziz equal to negative sign of theater differential of theater. And so we have that integral of sine of theater or we can, brother. Right? The internal We were calculating ahead. That is dangerous Off court of one plus u squared over by you and the integral off square root of one plus you square you better about you. Differential of you is equal to into a love sign of theater differential theories. Negative differential of Z. You bite by C square times one minus C square. Now we put this negative sign here inside these factor. So it&#x27;s a sequel to the Inderal of Differential of Z, divided by C squared times C square minus one and these can be transformed into the ah difference of two fractions is equal to the intro off Juan, divided by Z square minus one minus wanted. Bye bye. See? Square differential, is it? If we make these calculations, this difference of fractions here it is population. Here we get just this one here that is. Want about about six square minus one minus one over Betsy Square is equal to want about it by sea score times C squared minus one. And so we can have separate these into two into girls. So is equal to that Inderal off differential of Z divided by um C square minus Wong minus the interval of differential of C, develop Assi Square. And to calculate this internal, we made another to information partial fractions. Here this sequel to the Inter Love. Uh, when have one have times Juan divided by C minus one minus wanted. But I see both one made thesis calculation. Here we get Just wonder what? That C square minus one. That is because C squared minus one. Can we factor out as C both one time C minus one. Then we can ride thes expression here. And this intra here is second interval here is equal to the Inderal off. Sorry. Here is minus intervals e to the negative to differential of C. We can separate these in drill here. So it&#x27;s one have times internal of differential of C divided by seed minus one. You men at minus the control of differential Z divided by C plus juan a minus See to the negative two plus one abided by native to plus one and no, this is able to 1/2 times thes internal here is equal to the natural algorithm of the answer The value of CE minus one minus the natural lottery them off the absolute value of Z plus one. And here we find the denominator is negative. Two plus one is negative one with the negative year we find bless and wanted. But by Z because he see to the native one, we can say this is plus a constant of integration without any broken here. And this is equal then applying the bar properties of the natural logarithms. This is 1/2 times and that&#x27;s delivery them up the absolute body of C minus wand. But if I see plus one plus one of busy and now see we remember the sequel to co sign of theater. So we put that here. Then this is this is able to one health times and that&#x27;s delivery them up. Yes, for the body of go sign in theater minus wanted by the very co sign of theater Bless one plus one divided by go sign of the theater. But see, you know to know these expression co sign of theater in terms of you began to use a right triangle here where we put the angle theater and we taking to cap that you see sequel to tensions of theater here. So tangent of theater is equal to you. And to have these relationship here in these a right triangle we can put the opposite side to think I go to you and the adjacent site of three Goto want. So the eyewitness going to be square in the one place he square and then the co sign of theater, which is the jury metric function. We mean this case is equal to the adjacent side. The theatre that is wanted by the by the I partners that is the square root off one plus you square. And so now we can ride the expression here, which is the inter a Love the Inderal off the square root of one plus e square The but about you is financial of you is equal to ah 1/2 times and natural rhythm of the absolute value off um, co sign a theater minus one that is one divided by the square root of one plus you square. Maybe we can write it Another way to see clearly. So is one divided by the square root the one policy square that he&#x27;s go sign of theater here, here minus one. So minus one and one divided by the square root of Want Lucy Square plus one plus one. Better we go signs that he&#x27;s a reciprocal off vehicle sign of Syria, very square root of one policy square, the sea. And now, um, if we do the calculations inside the absolute values this equal to 1/2 times and that slavery them of the answer the value off one minus square root of one plus e square and way we&#x27;re going to have these divided by the square root of one plus you square. But because we will have the same denominator in this expression here, we can cancel out. So in and of being this one plus square root of one Policy Square gets because both denominators cancel out you square classic square root of one policy square. So what we have found here, we&#x27;ll see. He&#x27;s, uh, the central here and now we know that the original integral we were calculated that he&#x27;s thesis one here. It&#x27;s equal to negative, this one we have just calculated so we can say they know that the injury roll off the square root of one plus e to the negative to time sex differential of X is equal to negative and 1/2 times natural rhythm of the true value of one minus square root of one plus e square abided by Juan plus square root of one plus e square minus the square root of one plus you square plus see another Boston&#x27;s. And now we, uh, right this in terms off eggs, using the relation you is equal to e to the navy vase. So we go here. This is negative. 1/2 dozen artillery them of the value off one minus square root of one plus E to the negative X square that is e to the negative to 10 6 divided by one plus the square root off one plus E to the negative to 10 6 minus the square root of one plus e to the negative to tempt sex. Let&#x27;s see, this is the definite integral we can use another properties off the logarithms here to put this negative sign here as his opponents of the argument of the natural Laurie them. So we can say that this is sequel, or we can say that this implies that here implies that integral of the square root of one plus e to the negative to tempt sex differential of X Sequel to one have natural ovary them off the absolute value off. So we&#x27;re going to invert this fraction here. So it&#x27;s one plus. Is Claire route off one plus E to the negative to 10 6 divided by one minus were route of one plus E to the negative to 10 6 minus square root of one plus E to the negative to 10 6 Plus he And so, uh, now the lens of the church You see definite, definite integral from 0 to 1 of these functions this fresh in here. So the angel from 0 to 1 of the square root of one plus e to the negative to 10 60 French of x Sequel to, um uh, These expression one have times As for the freedom of the value of one plus square root of one plus e to the negative to tempt sex divided by one minus square root of one plus key to a native to 10 sex minus square root of one plus e to the negative to them six plus c. We can get rid of the sea because it&#x27;s going to be Cancel out his revelation here. Yeah, these evil waited between zero and one. And this means that this symbol to weave away discretion at X equal one we get one have times and that we re them of the actual value off one blast square root of one plus e to the negative chur. What if I want minus square root of one plus e to native to minus this query it rude of one plus eater. Native to that, that is expression even waited at X equal one out here at X equals zero. We&#x27;re going to obstruct this at X equal. Seriously, it&#x27;s negative one have, um I think I did one have off the natural luxury. Them of the absolute value of one plus hex equals zero. We had heat&#x27;s ears. It&#x27;s always skirt of to divided by one minus square root of two. That&#x27;s a value. And plus, because we&#x27;re so obstructing the expression. So last the square root off to okay. Minus OK minus. And these these defendants person, we have here number. And if we want to know more or less, which numbered pieces. We made a computation here and this is separatism. It legal to one point 19 27 So this is the lens of the care of given by white waas, um one minus e to the negative x between zero and one.

Rakesh Kumar Sharma · 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.

Clarissa Noh · Answer

Hey, it&#x27;s Claire. So in your radio. So we&#x27;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&#x27;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.

Problem

Graph the curve and visually estimate its length.…

Problem 18 Medium Difficulty

Find the exact length of the curve.
$$y=1-e^{-x}, \quad 0 \leqslant x \leqslant 2$$

Answer

$=-\left[\left(\sqrt{1+e^{-4}}+\frac{1}{2} \ln \left|\frac{\sqrt{1+e^{-4}}-1}{\sqrt{1+e^{-4}+1}}\right|\right)-\left(\sqrt{2}+\frac{1}{2} \mu \left| \frac{\sqrt{2}-1}{\sqrt{2}+1}\right|\right)\right]$

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$=-\left[\left(\sqrt{1+e^{-4}}+\frac{1}{2} \ln \left|\frac{\sqrt{1+e^{-4}}-1}{\sqrt{1+e^{-4}+1}}\right|\right)-\left(\sqrt{2}+\frac{1}{2} \mu \left| \frac{\sqrt{2}-1}{\sqrt{2}+1}\right|\right)\right]$

Essential Calculus Early Transcendentals

Catherine R.

Heather Z.

Caleb E.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

James StewartEssential Calculus Early Transcendentals

Catherine R.

Heather Z.

Caleb E.

Joseph L.

James Stewart

Essential Calculus Early Transcendentals