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Find the length of the curve: 16) y = (9 - x2/3) 3/2 from x = ] tox= 27 4) 24 8C) 36D) 72...

Question

Find the length of the curve: 16) y = (9 - x2/3) 3/2 from x = ] tox= 27 4) 24 8C) 36D) 72

Find the length of the curve: 16) y = (9 - x2/3) 3/2 from x = ] tox= 27 4) 24 8 C) 36 D) 72



Answers

$$
\begin{array}{l}{\text { Multiple Choice Find the length of the curve described by }} \\ {y=\frac{2}{3} x^{3 / 2} \text { from } x=0 \text { to } x=8 . \quad \mathrm{}}\end{array}
$$
$$
\begin{array}{ll}{\text { (A) } \frac{26}{3}} & {\text { (B) } \frac{52}{3}} & {\text { (C) } \frac{512 \sqrt{2}}{15}}\end{array}
$$
(D) $\frac{512 \sqrt{2}}{15}+8$

$(\mathbf{E}) 96$

Okay, so we must calculate the arc length for this curve in this interval. So we will do this using the formula for arc length, which is here in green. And the first thing we need to do is compute the derivative of this function. That's fine. Do I. D. X. Personally use power rule? Sorry, 1/2 there we are. Then we have to chain rule it. So let's take the root of the inside. So a power will once again take the power down and subtract one To the -1 3rd. There you go. And so now since we need the derivative squared, I'll go ahead and just figure out or compute the square of this derivative down here. So first of all these fractions will cancel three hives and 2/3. And so the square we can distribute between these between the product. So this first part, the one half will just get canceled. So we'll have a nice four minus X to the two thirds and then X to the minus one third, multiply the power by two. Since we're squaring it Comes next to the -2/3 and then we can distribute This X to the -2/3 to the terms inside. So this simplifies nicely too four times X to the minus two thirds -1. Because you can see the experience will add to zero. So actually zero will just be one. And so now we can compute are integral From 1 to 8 to grow. And if we evaluate this really quickly we can see that the square is this for X the minus two thirds minus one. And then If we add one, the positive negative ones were just canceled. So we just had square root four X 2 -2/3. Close the root dx So the square root, well, cancel the two in our power here, You can think of the square as a 1/2. So we have four x. So -2/3 All to the one house. And so once again will distribute the square root between the product and so we rewrite are integral here, Screwed of four will just be too and squared of X to the minus two thirds will just be X to the minus one third dx. So now we can finish evaluating our integral. So we use the power rule for our integral. So this becomes three halves X two thirds. We added one and then divided by the power Evaluated from 1- eight. And so the two and 3 house will cancel. So The third root of eight is to let me just run it out. So we get two times three has eight To the 2/3 minus three halves, one two thirds. And so we can cancel the twos. This becomes three times eight to the one third is two squared is four -1 to any powers, just 1 -3, So this is simply nine.

So we have to find the arc length For this curve for this interval from 1-3. And so we will use our arc length formula here in green. So what we need is this DVD X. So let's go ahead and compute this derivative. We'll use the quotient rule which is here in red in case you do not remember. So let's go through it at the top times the bottom minus the herd of the bottom minus the top. Whenever you do this for all over the bottom squared. Okay, so let me produce this down a little bit. Okay, let's simplify it a little more. Take out of six. No, we can actually take a 18. So if we this becomes 18 x 34 -18, We get extra 4 -1. All over two X squared. And so now let's evaluate This inside part of the Integral 1st. Before we actually do integration because it should cancel nicely. So if we square this, I'll use an arrow. So it's not cool. So you want to square this and then add one. So to expand the fraction here. Yeah except the eight -2 X. The four. This one All over four X 4. And now I'm gonna give this a common denominator so I can add it into the fraction Works of the 4/4, 4. And so This becomes extra eight minus no plus four X. Before now because we add this for plus two X four, We have this negative 2-plus positive for it gives us positive too. Plus one. All over four X. to the floor. Now this top part can be factored is a square. You see extra four plus 1 all squared over forest before this is nice because inside are integral. We're gonna have to take the square root of this. And both the numerator and denominator are both in square forms forms that are easy to take the square root of. So let's set up our integral here from 1-3. Those are pounds square roots of this quantity here. So let's go ahead and just take the square. Right now this becomes Next to the 4-plus 1. All over two X squared. So I can get rid of the square root. So you get X to the four plus one. All over two X squared. Just make sure you got that right? Yeah. Dx Now to evaluate the integral will distribute the denominator. So we get x squared over two plus one half. Let me take the 1/2 out. Sorry, robert perhaps squared plus Next the -2 the X. And so now we'll just evaluate the integral with power rule. So this becomes next to the 3/3 minus X. to the one Over one so just one over X. Um Right Evaluation bar from 1 to 3 of course a parenthesis. So let's plug in and evaluate, We got 1/2 So be 9 -1 3rd. We get some more brackets minus one third minus one. So this is nine is 27/3, so we get 26/3. One third minus one is negative two thirds or subtracting, so becomes positive two thirds And then we get 28/3 times one half. This is comes out to 14 over three.

We will determine the length of the curve defined by this equation Between the bounds y equals one and Michael's three. And we will do this using the length of a curve formula here in green. And so to find the derivative you need, you can do implicit differentiation, but I'm just gonna go ahead and solve for X here. So first we divide, Let's just do it all at once. We're gonna divide by a 30 y cube. So we get Next -Y. The 5th over 30 goes one over two, I cubed and then going to move the way 5th terms that was signed. So you're equals plus. There we are. So now we can differentiate this with respect to why? And so again, What are the four over six minus one over, sorry, 3/2. Two. Why? To the four? There we go. And so now we can square this and and one. So first let's square it. It's supposed to be why the eight 4 36 minus two times the product of the two terms. So that will be one house. That's one half or 3/2 divided by six Will be 1 4th Times two is 1 half. And then the last term squared here. And when we add one to this, you can see that Because we'd add one here to get the term underneath the radical and are integral. This one half will go from positive from negative to positive. And we can see that this factors in the same way as our derivative. Just with a positive science. So we can now write out our integral here. So this will be an integral from 1-3. And so we said that When we add one, let me write it out out here. So this will be our term underneath and underneath the radical inside of our integral. So be What are the four or 6 Plus three over too wide. The four. This will be all squared And you can file this out to see, you'll get the same as this party member with the plus 1/2. And this will cancel with our radical here. So inside the parentheses, this is what we have inside are integral. So You have one of the four over six plus 3/24. This is all dy and so now we can just evaluate are integral Social B. What are the five over 24? Sorry over 30 minus one over to 1/3, Evaluated from 1- three. And so now let's do some plugging in. 3 to the 5th is 243. We can take away three from the bottom so we get 81/10 -3, Cubed is 27. So this is 1/54, and then this is minus 1/30 minus one half, Putting the one here. And if you plug this into, or if you find common denominators, and this was simplified to 1154 Cups, 11 54 over 1 35.

So in this question, we know that since the arc length of the portion of the graph of back to backs on our interval, 80 is gonna be given by. So they are. I'm just gonna be given by this, and I will. So this is basically the formula we're gonna be using. And it's an axe squared on the D. X on the end. So this is not for me either. We're gonna be using. And now we started playing when we know we know that our maybe you're gonna be zero in eight. You know that from a question. And then we have one plus x squared over four. And then this is Times GX. Uh huh. So now, um, I want to be paid. Sure. We can make this for you. Want half? There will be 80 times swearing a four last ax squared times, dear. And next we're gonna make this be you want, huh? Times X over too. I was a swear word for a pleasant experience. Plus, for over two years, which we spent two not full log of axe, plus no. In the square root of four. What's expert expert? Remember that is a dancer. Final beige. Um, Now we're just gonna play in our exercise, so we have 1/2 times over. Dude, I'm just a spare room. Four. It's gonna be This would be a square. So 64 lost to natural log of axe plastic. That the same thing that we had for four, But it's weird. So we understand. For 64 mice, 1/2 this is too now on to one of two on two. All right, so now we could just point this big thing of torque on Twitter. Think this, like, 68. This would just be on too. And we get 18.5 87 is our final answer, This question.


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