find-the-exact-arc-length-of-the-curve-over-the-interval-xfrac13lefty22right3-2-text-from-y0-text-to

Question

Find the exact arc length of the curve over the interval.
$$
x=\frac{1}{3}\left(y^{2}+2ight)^{3 / 2} 	ext { from } y=0 	ext { to } y=1
$$

R M · Accepted Answer

Brooklyn number four, The lens is equal to take, ranging from 0 to 1 one plus boys where Weisberg plus two de Roy were injured. Age boy is equal to, uh, Why little or voice bread less. Um, and this can be simplified to and a secret to integration from 01 off, one over to spread review one plus U u plus to the you on this immigration for that is equal to invigoration from beautiful one. Oh, you plus 1/2 square of you, which is equal to integration from 01 Beautiful. But who&#x27;s griddled off you plus aggression from your one on one over spurt of you you, which is equal to here u is equal to voice where? So you know why. Why so ah do boy is equal to 1/2. Uh, split you, Dean. So the integration for this for Melisse Ah, you or speak over to over 3/2. Plus, uh, you were her over, huh? From 0 to 1. Dr Those of institution. But why? Ah, but for you is equal twice. Where? So it&#x27;s a big toe. Um, why? For 3/3 bus? Uh, why over her from 0 to 1. Do you think well after substitutions, apartment accidents were over

R M · Answer

problem Number three. The list is equal to a declaration from A to B. It&#x27;s were off one plus a dash of beds where eso figuration from little ones. Square root of one plus 9/2 square foot Opec&#x27;s. It&#x27;s where yet, which is equal to a bird with a teenager from beautiful one integration or both. Plus 81 over. Or it&#x27;s, uh, the XT with his equal toe, given that paperbacks is able to three X being over minus one. So if bash off X Sequent and nine ever to square Kovacs eso here and is equal toe for over 81 a little of you. The creation from Beautiful what which is equal to over 81 times for three over to over 3/2 from 0 to 1, is equal to, uh, eight. It is equipped to okay, which is equal to ah eight. A over two for three 81 x over or plus one on city over to 101 which is April off the substitution by one and zero. The bundle answer would be one or but for three 85 square off 85 minus honey State

R M · Answer

problem of about five He was over. Es. They were due to over three X or negative one off three j deeper to over three X forward. Ah, differentiation function. So any secret integration from one, 28 Ah, square one plus for over nine. It&#x27;s for over three. Yes, you think with instigation from 1 to 8 of what? Gus who were three? Expert three. Where yet which is equal to upto simply education. Ah is equal to integration from 1 to 8. Ah, expert off nine X over. Over three plus four over a squared off nine. It&#x27;s our approval. Three. Yes, it&#x27;s a sequence investigation from 1 to 8. Uh oh. We&#x27;re square off nine export over three plus four over. Export over three. Huh? Yet which is equal to 1/3. Immigration from 1 to 8. We&#x27;re open nine x over three plus more over X power. Wonderful. Three. The x eso the integration for that is able to 1/27 nine x for over three loss More old over 3/2. Or what? From one wait upto service fusion. We can get that final answer is 1/20 seconds. 80 We&#x27;re off then. My nuts. Europeans

Clarissa Noh · Answer

it&#x27;s clear. So when you read here so we have f of facts is equal to two X to the three house power. So that gives us a derivative. The three ex So 1/2 power one plus that they&#x27;re a bit of square. Gives us one plus nine x someone we plug it in to our bowling formula. We&#x27;re just gonna change acts to t you get the integral from one to x square of one plus 90 DT, we&#x27;re gonna make you be equal to one plus 90. So nine e t is equal to do you so d t is equal to do you divide it by nine. This changes the limits of integration. So it goes from one to arks and it becomes ton. Since we&#x27;re adding nine and one plus nine x you of 1/2 power, do you over nine. This is equal to one over nine turns 2/3 power 2/3 comes you to the 3/2 power. This is from 10 to one plus nine x. This gives us two times one plus nine x the three house power minus 20 square root of 10 over 27

Rakesh Kumar Sharma · Answer

fully so we have to let the function Oh, just you in this half off exit goes through who exercises worked. You OK? And you have to calculate the lead from X. We&#x27;re very possible. So that stop start from one and we keep doubling this excess. Excellent. So let&#x27;s just different your disgust or diverse nation fester on best off X will be pursue. Who knew you? Do you know access to Parliament? Who do we cancel out the first of your DVR? Vernon&#x27;s ex team plexus apartment Where? So for the let off call. So angry because you integration went to X. Okay, the lorry video review one and you see excessively Then when we have a business. So the upper limited almost people because of illegal. Hey, you know one witness. So this is he&#x27;s So now you get this life you need to get This is the integration with the super essentially, actually, one plus nine next day. Super hot. When my you re x indication within weeks. Okay, so let&#x27;s indicated So the indication you&#x27;re going to hear this one. Bless Nynex&#x27;s Jabbar. Chief, I do. Okay. Do I never see why do so. I think it does in the new murder. So they become who by three. And we have to be ready by the coalition or excessive. So there isn&#x27;t OK. And then Laurie, with this woman, A preliminary accident report here now, before this. And if I select for the simplified to put the upper limit so we could buy team and we can take my old turn, its with surgery become actually. Okay, We think that Soran do one by nine. Okay, So Parliament people that extra this remain as in so it really is in Chile. Well, bless my next issue. She might do OK. My yes, Lord, we have a good one. So you don&#x27;t know. 92 129%. So that pretenders to Bob we were you. So before this in the fight, the length off, love you were doing this who way would be something but a worthless my next days about do you do minus Oh, and by that Oh, yeah, that would be and good. Okay, So that

Kami Dupree · Answer

right this question. We&#x27;re looking for the arc length of y equals X to the three halves Power on the interval from 1 to 2. We have hair and read the arc length formula as represented on page for 68 and so is part of that formula. We know we need to find the derivative of the function, and then we need to square it. So we&#x27;re gonna go ahead over here and find the derivative of ah X to the three halves. That, of course, is going to be three halves X to the 1/2 power. Three halves minus one will be 1/2. And then, of course, we need to square that. So why Prime squared will be three halves to the next. The 1/2 square or 9/4 and then remember except one have powers like the square root of X plus the square root squared, which just means we&#x27;re gonna get X so we can come now and say, theater girl that we&#x27;re looking for for the arc length is the integral from 1 to 2 of the square root of one plus 9/4 x d x, which is deceptively simple looking but we are gonna rewrite this instead of a square root. It&#x27;s a little easier to integrate when we think of that as a power. So we&#x27;ll think of that as a 1/2 power instead of a square root that will allow us to rely on the power rule as we do our inspiration. Now there is a little trick. Here are a little hiccup. It&#x27;s going to come up as we do this integration. It&#x27;s gonna seem easy enough, right? We&#x27;re gonna take this function here and say It&#x27;s too the 1/2 power. So as we integrate that, we would then say We now want to take one plus 9/4 and we would now raise it to the three halves power and then divide by that new power. But we can rewrite that, Of course, as 2/3 times right, divided by three halves is the same as times in our multiplied by 2/3. So well, right, that is one plus 9/4 too. The 1/2 multiplied by 2/3 and it might be tempting to think that we&#x27;re finished with this integral. However, we have a small little problem. I should be a three there we have a small little problem that may get overlooked if you&#x27;re not careful. And that problem here has to do with the fact that let&#x27;s just take well we have here. And let&#x27;s say that we&#x27;re done with our integral we were done with are integral. Let&#x27;s just do a little side trip over here and ask ourselves, What is the derivative right? The derivative of this. If we&#x27;ve got the right integral, the derivative of this should give us one plus nine Force X to the 1/2. So if we take the derivative of 2/3 times one plus 9/4 X to the three X and then all that to the three halves, then we would then take three halves times 2/3. That&#x27;s great, cause that&#x27;s one, and we would have one plus 9/4 X to the one less power. So that would be, too. The 1/2 which which looks good until you realize that is part of the derivative. You would also need to take the derivative of what&#x27;s inside of the argument there, and so the full derivative here would also be multiplied by the derivative of one plus 9/4 ex, which would be nine force. That means as it stands, if we leave our integral if this way, we haven&#x27;t really got the integral of this function that we initially started with because our current derivative has an extra 9/4 in it that we need to account for. So the way that we&#x27;re gonna account for that basically is to say, actually are integral, then needs to be able to account for the fact that we would multiply by 9/4. And that means that we&#x27;re going to say we would divide by nine force. In other words, if we were now to take this derivative. But this thing that we started with had a 9/4 down here that will be times by nine Force and divide by nine Force, those would divide out. So the trick here is that you&#x27;ve got a little bit of integration within the integration that you need to take care of. And so, in actuality, here are integral becomes again. 2/3 divided by 94 So I&#x27;m gonna call that 2/3 times 4/9 and one plus 9/4. Who should have an X in here? Mine fourth ex to the three house. And of course, we&#x27;re evaluating that between exes one and X is too. So make sure you keep this in mind, because that is a little bit of a trick when the thing that you&#x27;re integrated so we&#x27;re integrating and 1/2 we&#x27;re square root function. But within that, we also have another function. So we&#x27;ve gotta account for that. All right, so I&#x27;m gonna clean up, are integral. So our arc length is basically now the integration is complete. We have eight 27th times, one plus 9/4 x to the three halves, and we&#x27;re evaluating that between exes one and X is too. All right, so we need Thio do the evaluation here to do the evaluation. I&#x27;m actually gonna add another page here. Let&#x27;s go over to page two and I&#x27;ll rewrite what we&#x27;re doing here. So we&#x27;ve got a 27th times and I&#x27;m just going to switch the order here 9/4 x plus one to the three halves and we want between one and two. All right, so let&#x27;s, uh, duck to in. That&#x27;s going to give us a 27th tying 9/4 times two plus one all to the three halves. And then, of course, we will subtract from that. And when we don&#x27;t want into the integration, we&#x27;ll have 8 27th times 9/4 times one plus one to the three house. Now we&#x27;ve got some algebraic manipulation that&#x27;s going to get a little tedious, but certainly doable. 8 27 THS here and 9/4 time&#x27;s, too. That&#x27;s 18 4th plus one to the three halves. And, uh, just while I&#x27;m here, I&#x27;m gonna go ahead and go one more step with that and say, Add these fractions. One is 4/4. So this is 22 4th to the three halves and then we&#x27;ll subtract here and let&#x27;s see. Over here we&#x27;re gonna have a 27th times 9/4 plus one to the three halves, which again this one will be 4/4. So we call this a 27th times 9 13 will be 13 9 force plus the 4/4 eso. We&#x27;ll call that 13 4th to the three halves Now, even though I could do some simplifying here, I&#x27;m not going Thio just because the algebra is gonna work out a little better. Well, it won&#x27;t matter if you do or don&#x27;t do it. But remember that the three halves power can be written as a square root and acute. So we&#x27;re gonna call this 8 27 times the square root of 22 cubed times four divided by four cubed. And over here, we&#x27;re gonna do something similar. 8 27th times the square root of 13 cubed, divided by four cube. So that&#x27;s three halves. Power is just the square root of a cube. But I&#x27;m gonna even break that cube down again, and it&#x27;s gonna seem kind of silly them to do this. But I&#x27;m gonna say it&#x27;s 22 squared times 22. That&#x27;s still 22 cubits, four squared times four. Now, my seem kind of you might be scratching your head as to why I would do that, but I&#x27;m doing that in part because I have a square root and a squared so square on a square gonna end up canceling each other out, which means, you know, for example, if I had the square root of three squared, then I can just say that that&#x27;s three. So if I have the square root of 22 squared and square to four square. I can say this is a 27th times 22 4th I can pull those guys out of the square root and then multiply that by still the square root of 22 4th And similarly, I&#x27;ll do the same over here. A 27th kinds will be able to pull 13 4th out because there&#x27;s a squared in there, right? This is 13 square times 13 and four square times four. And then if I take the square root of the squares, um, I&#x27;m gonna be able to pull the 13 4th out. All right, so now let&#x27;s take this mess up here and finish off the algebra. We&#x27;ve got 8 27 22 4th Of course, the eight will be divisible by the Force, so we&#x27;ll call that a two. And this is 44 27th times. I&#x27;m gonna rewrite this square root of 20 to force as square to 22 on the square to four. Remember, we can split the fraction into two square roots and then similarly here, the eight and the four gonna divide out. So we&#x27;ll call this 26 27 THS times the square root of 13 over the swear before. Now you can see why I want to do that. Because the square before I can evaluate, right, this is gonna be too. And this is gonna be too. So that&#x27;s nice, because that, too, is gonna divide out here with the 44. So I can say, Let&#x27;s divide that out and make that 22. That gives us here on the left 22 square roots of 22 over 27 and then similarly on the right. This, too, will reduce that to 13 and we&#x27;ll end up with minus 13 roots of 13 over 27 which we can leave in that format. That would be the arc length. Or we can say we&#x27;ve got a 1 27th is a common factor there and that pulling that out will make this just look a little bit nicer. It&#x27;s 20 to 20 times 4 to 22 13 times the square of 13. So either of these they&#x27;re both essentially the storm

Problem

Find the exact arc length of the curve over the i…

Problem 4 Medium Difficulty

Find the exact arc length of the curve over the interval.
$$
x=\frac{1}{3}\left(y^{2}+2\right)^{3 / 2} \text { from } y=0 \text { to } y=1
$$

Answer

$L=\frac{4}{3}$

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$L=\frac{4}{3}$

Calculus Early Transcendentals

Grace H.

Anna Marie V.

Kayleah T.

Michael J.

Integration Review - Intro

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Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

Grace H.

Anna Marie V.

Kayleah T.

Michael J.

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Calculus Early Transcendentals