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Question 22 ptsThe volume of the solid generated when the region between the erphs J = Jd } Ovel the: intewval [W] is revolved about the axis IS:2T3Ix33 TC...

Question

Question 22 ptsThe volume of the solid generated when the region between the erphs J = Jd } Ovel the: intewval [W] is revolved about the axis IS:2T3Ix33 TC

Question 2 2 pts The volume of the solid generated when the region between the erphs J = Jd } Ovel the: intewval [W] is revolved about the axis IS: 2T 3 Ix 3 3 TC



Answers

Sketch the hypocycloid $x^{2 / 3}+y^{2 / 3}=1$ and find the volume of the solid obtained by revolving it about the $x$ -axis.

Okay, So for this problem, we have the function. Why equals one over X And then we're, um where X is going to be between two. And B. Um, actually, it should be the other way around because B is actually going to be less than two. So beat it too. And then, of course, this is also gonna be y in zero. So what we want to dio is I want to go ahead and set up the volume Thio Equal pi and then the integral from B to to. And then what I want to dio is do my function. So it's gonna be one over X and then squared. And so then what I want to do is it's gonna end up being yeah, pie. And then from Bt two of one over X squared, okay. And then DX. So then this is going to end up being pie and then times one negative one over x from two to be So this is going to end up being pi times negative 1/2 plus one Overbey. So I can say that this is going to be pi Times one Overbey minus 1/2. Now what? I also know that this volume is going to equal three. So what I want to do is I want to find out what value of B is going to give me this. So I'm gonna go ahead and distribute out this pie so pi over B minus pi over two equals three. So then I can set this to be, um yeah, I could go ahead. Do pie Overbey equals three plus pi over two. So, in other words, this is going to be pi over b equals six plus pi over to, and then I can go ahead and, um, divide out the pie so it's going to be, um, won over b equals six plus pi over two pi. And then all I want to do is just do the reciprocal of both sides, so be would equal to pi over six plus pi.

In this problem. We want Teoh find the volume of the solid generated by. So why is equal? Teoh expired over nine minus X squared on the interval. So X is in zero. That's too and it's rotated about the X excess. This is some kind of solid. Okay, so we're gonna have the integral from 0 to 2 of high r squared so high, What's are ours wise a y squared pie y squared TX So it's gonna be 0 to 2 high times X to the fourth or I guess, X squared square How I want to write this You do Excellent. Fourth over, Um, stop. So nine minus X squared squared theatrics. So we wanted to you partial fraction decomposition. But we're in a little bit of a pickle because here right, we don't have our denominator has the same order is the numerator so we can do one of two things. Either we can multiply out nine minus x squared, squared and then do long division sounds terrible or we can dio a little put take So I wanna subtract nine x squared and then add it back. You might be like what does that even dio. Well, if we have X to the fourth minus nine x squared, we can factor out on X squared and we'll get X squared times. What's this gonna be? It's it's gonna be, um, X squared minus nine. Great. So I'll split this fractional supposed take the pilot front because we don't really need it in the integral. So we'll split this fraction up into two pieces, So I'm gonna have x fourth minus nine x squared over nine minus x squared. Quantity squared. And I have this other term, so I'm gonna have plus nine x squared over nine minus x squared. Quantity squared. So first, let's look at this first traction. So what's this gonna be? It's gonna peck squared over. It's really mean negative. Excellent. So we got X squared times X squared minus nine over nine, minus X squared quantities. Word. So this big animals flight by a negative so pulled negative out, that is gonna be a power. That's where I'm negative. X squared over nine, minus x squared. Now, this piece, this piece we're gonna do, um, partial traction and decomposition on. Let's finish up with this piece. So here again, we're kind of start. We can't use partial fraction decomposition, but we can do another protect. So we nuke plus nine minus nine. And so that will be we're gonna get, um So if you look at this piece of August 1 and this piece is gonna be negative 9/9 minus x squared. Make that a plus and plus, and we could make the bottom X squared minus nine. Now, I can use partial fraction and decomposition on this because I have my numerator is nine and my denominators X squared minus nine. So the denominator has higher order. Okay, cool. So before we do any of this nonsense, let's go ahead and rewrite this. We're gonna have high times one plus nine over X minus three X plus three waas nine x squared over X minus Uh, X minus three Squared X plus three squared. When I did was I factored out that denominator and we gonna dx okay, so we gotta dio partial fraction decomposition twice. So it's look at, um we'll just call this Q one and Q tip proportions. Okay, so for Q one, right, bring at nine over X minus three times x plus three is gonna Who'll a over X minus three plus B over X plus tree roots repeated so we can go ahead and say So we use the heavy side cover of methods so actually pulls three and X equals negative three of the roots of those factors. So a is unequal 9/3 plus three that's unequal. Three Halfs and B is gonna be nine over negative three minus three equals negative three House. All right, so those weren't bad. Let's go look at pew to Future is gonna be less one. We get nine x squared over X minus three Swear Times X plus three squared is equal to a over X minus three plus B over extra industry squared plus C over X plus three plus D over X plus three. Squared. Right. So since we have so many repeated roots, we're gonna have to multiply the denominators anyways, So let's go ahead and just do that. Now, instead of trying to play with the heavy side cover, so have nine. X squared is equal. Teoh eight times X minus three times x squared less six x plus nine plus B times X squared plus six x plus nine plus c times X plus three times X squared minus six X plus nine plus de times X squared, minus six X plus nine. All right, So for these ones, I was just multiplying out X minus three quantity squared. And, uh so I guess I was explosive Requited square and the other one's X minus three party squared. OK, so we got Ah, a little more motivation and then grouping and I will be there, so I'm just gonna look at. So what is X minus three times X squared plus six experts use I get X. I was using her color make sleazier. So when I x cubed plus six x squared plus nine acts and then the minus Treacher So minus three X squared minus 18 ax minus 27. So what sacking me? So we have X cubed will really have her ex court. We have plus three x squared. Then we have minus nine x minus 27. That seems right. Most thing about that So we have X squared and we get three X for X. We get negative nine and then FBI looks all right and then we want to multiply this one out tubes were X cubed minus six X squared plus nine X plus three X squared minus 18 X plus 27. That's X cute. Minus three X word minus nine X plus 27. Yes, I'm seat. OK, so ready. We're ready to go. We got a times this, and we just got out. We want a group by factors, Becks, we get nine. X squared is equal to So what's times excuse? We have a plus C X cubes. And then what's times X squared. So we got a bunch of stuff, so there is gonna be three a plus B minus three C plus de and that's times X squared. So then, for acts, there is a god of nine A, uh, plus six B minus nine C plus. What? So, um, did you do worse? T o plus nine D? Well, plus minus 60 minus 60. Okay, that was times X and finally constant. So we have nine are Oh, my gosh, my ass. 27 a plus nine B plus 27 c plus nine d and those were just constant. Okay, so we can play around with this a little bit to make it less terrible. So are we have zero equals a plus C and nine equals three a plus B minus three C plus D as zero equals negative. Nine a close six B minus nine C Final 60 and zero equals negative. 27 a plus nine B plus 27 C plus 90. All right, so we just have to solve the system of linear equations, and then I will get us infractions that we can play with. So let's go ahead and put this in matrix form. So it's a little quicker to solve. Where is my scroll bar? It just doesn't exist. Okay, Um really? Here we go. Okay. So, uh, what's Matrix form? So we get 1010 031 negative. Three 19 This will be my augmented form. Yeah, negative. Nine six. Negative. Nine Negative. 60 And last we have, uh, negative. 27 9 27 90 Okay, so we got some good reproductions do here, so we have negative three row one. Plus wrote to replaces. Wrote to We have nine row one plus road three replaces row three and 27. Row one plus row for places. Road for so 101000 one negative. Six 19 060 Thinking of 60 and 09 That's gonna be 54 90 Okay, uh, next. So we get a negative six row two, plus row three. Replaces. Row three and negative. Nine. Row two, plus row, four places. Row for so a lot. The old stuff remains the same, but so this will be 36 this will be negative. 12. And so native. Six times nine is, uh, negative. 54. And so zero zero 54 0 And so negative nine from science. That's negative. 81. Okay, so from here, we can solve it with back substitution. So we get C equals, uh, so that's gonna be negative. 81/54 which is gonna be negative. Nine over sex, which is going to be negative three halfs. So from here, we can get deep. So we get, um, negative. Three halfs times six squared. So, what is that gonna be? So, uh, negative. Three times 36 over to minus 12. D equals negative. 54. So half of 36 is 18 Three times 18 is 54. So that means he is zero next. So we get negative. Three halfs minus six. That's all right. So we get, uh, be minus six times negative three. Halfs is equal to nine. So what is that gonna be? So this is gonna be B plus nine equals nine b equals zero. All right, so then for a we get a minus three half zero a equals three hops. All right, there we go. We got A and C. So B and E were all zero. Okay, so from here. Okay, um, we can rewrite the interval, so get one. Where is all that stuff? So one plus. So what was the original A and B. So we get three halfs. We get three, um, three halfs, one over X minus three, uh, minus three halfs. One over X plus three. And then from the second fraction, we have So a waas three half. So plus three halfs One over X minus three plus Sorry. Minus three haps one over X plus three ds so we can combine those. We have a pie in front, so we're gonna have high one times. Three over X minus three on the integral of goes from 0 to 2. Uh, minus three times one over X plus, three X. All right, we're ready to integrate. So we're gonna get pi Times X plus three Ellen. Absolute value of X minus three minus three. Ellen. Absolute value of X plus three evaluated from zero to. I was gonna cry times so we can just evaluate the extent house that's gonna be too. So plus three Ellen of one, which is zero finest three Ellen of five plus three. Ellen of three, minus three. Ellen of three. I was gonna be pi to pie minus three Ellen of five because the we're gonna cancel at zero and yet

Normally I'd be scared of this problem but with the fact that we're evolving around the y axis makes us a somewhat doable problem because if you look at the problem uh temporary right this as Y instead of F of X. And we have three over one plus X squared. Uh If you were to solve for X. So what I would do is kind of like cross multiply uh make it one plus X squared to multiply over and then divide the Y to the right side And then subtract one to get rid of that one and then square root. Um The reason why that would be your benefit is because when you revolve, the function is going to be high times the integral of that square root piece. Three over y minus one quantity squared. In terms of why? Now, what we also need to consider is it said from x equals 02, X equals three. So what we need to do is figure out what the why values are. Okay, well if x equals zero, uh we would end up through a three divided by one, which would be three and if X equals three, well three squared is nine, one plus, nine is 10. So we have 3/10 hopefully into that correctly. Yeah. Um And the other piece I really like about this is the square root in that square would cancel. So when you're asked to do the integral of this we would have three times natural log of y minus why in the drift of of this should give you That piece up there from 3/10 to 3. And so what you need to do next is plugging those bounds so pie uh times three natural log of three minus three. And then minus as you plug in the other bounds, three Natural log of three tents become plus because you have to distribute that that minus two, two, that's only 3/10 there. Uh And this should be a perfect answer. It's just a matter of how much you want to simplify before you circle it and move on.

This problem. We are asked to find your boy. You wouldn't get a fart Revolving the region. This shade of Yoo Jin Barone dances. And this is the region between this term to over skirting board. Uh, the XX is so more zeroes than 13 So we know that in order to find this, is there no holes or shells? We could use this record and it tells us that William, we'll be able to buy time function. Um, for me, her off squared terms be one. Now the levels are from zero to be again. We are integrating the ones you want we have on and our function as to whiskered of my pulse. Also, if you were to take spirit of back, he would have four incinerator and one plus one in the denominator. You? What do you want? Four planes is the constant and you don't have the triangle One over one plus 20. Life is just nature Law off my plus one. It went, This is 03 So we would find the war. The rest or fine National level four minus next. Love off one where? Mister Zero. So the answer is four fine law four or since four is to spread, we could take this over and lunch is the most play. It would be eight. Hire a model, too.


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