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IWI6.2v2: Problem 9PevUpNextPI Find the volume of the solid generated by revolving the described region about the given axis:FThe region in the first quadrant enclo...

Question

IWI6.2v2: Problem 9PevUpNextPI Find the volume of the solid generated by revolving the described region about the given axis:FThe region in the first quadrant enclosed by the curves y 4x and y = 22 rotated about:lahThe T-axis: Answer: 704pi5b: The y-axis: Answer: 0Note: You can earn partial credit on this problemPreview AngwermBubmit AnpwenIYou have attempted this problem 8 times Nour overall recorded scorellshonal Iou have unlimited attempts remmainingnEmaj insbructor

IWI6.2v2: Problem 9 Pev Up Next PI Find the volume of the solid generated by revolving the described region about the given axis: FThe region in the first quadrant enclosed by the curves y 4x and y = 22 rotated about: lahThe T-axis: Answer: 704pi5 b: The y-axis: Answer: 0 Note: You can earn partial credit on this problem Preview Angwerm Bubmit Anpwen IYou have attempted this problem 8 times Nour overall recorded scorellshonal Iou have unlimited attempts remmainingn Emaj insbructor



Answers

Find the volume of the solid generated by revolving the region
in the first quadrant bounded by $y=x^{3}$ and $y=4 x$ about
(a) the $x$ -axis$
(b) the line $y=8 .

What questions? So this problem has two bucks. But clean up the problem we have you buy your ticket. Did I do? By who? Caustic steps. This is gonna end sine X minus one in blue. Senator Dio is your picks, That is yes. On Sex Limited here to Pilot. This is equal bye plus costs by like minus D plus one buy into one last video minus slim. So this is equal to buy weapons now, if you want. Yeah, it is Dubai and do by one minus one. This is given as good bye and bye bye to minors. This Kansas becomes five minutes by This is even as bye bye. This is Dancer for part of the problem. Now we have beyond the problem. Wait, We have to find William. The other program is the Iwas by Theo. Do I do? Why? Thanks into graphics D yes. This is good enough. So you know we're going to find it. Uh, delivered just by my minus six and the graphics gives This is giving us 500 minus six. Do you love gone 60 years minus? You have one day itself. I would be minus six is no on 60th or he's just given us. Why do minus six? You do sign X minus and it is no minus one. You do sign. It's the ex. This is good. My mind says Sinus. Bless you. No. Seven days. This is a mess. I'm a minus six into Senate minus six. No, this is giving it minus caused by a minus. Buy food minus Gino. Did you know my ass? Oh, you know, really? This is your dinner's my mess. Judo my nuts, man. Is it Dennis? Well, therefore bowling waas, you'll find one. This is a problem and we have answers from people.

56 decibels to fine a volume of the region. William, off a solar which is generated by revolving the region which is bonded by eccentrics with a graph of exciting mix between zero plays like this. Uh, the media just wandered by this exciting next X axis. Onda Uh, this, uh, but the next act zero in place. So this is the required region on this is rotated first about why exes in the signable executed by Let's start with party. And this is what you're gonna vote by access. So I think the shell matter will be more comfortable here. So we take a small anger over here like this. It's just me in Australia's, which is basically it's and eight basically. Why So the one you will be by X Why we have to integrate this X. We have to win. Degrade this, but respect to X within the limits. See her? We have the value of y as x sine x. So that's replaced us. Has become group I X excitements. That will be a square say necks dx dy greatest receiver. So we hear me after use integration by parts of the sparkle. Just are baby means old suit First Timers Editors. Integration of second term will be transpose X medicine to go off the finishers. Differentiation of Customs two weeks indeed. Jealous minus, of course. Six. Unisys for the indicator, so this becomes stupid minus X a square close six plus two Digger No X Koszics Deitz once again into go by integration by parts first term second term and let's open up the brackets as well. So this becomes minus. In fact, let's keep my to buy outside on. Let's open the back. It's in the next time, Uh, this will become first time either prison people of second on the ST Nick's minus integral of defense Station off first Thomas X. A different station off Western, which is X is one integral integration off course will be saying, and they should be once again integrated. So this becomes do you buy minus X squared for six Bless you. X signings, minus indication of saying remain a Scots simplicity. All sex. And here now we have to put elements See you. So who pays outside cause to place for space menace ones off minus by square minus one, ST Bay zero and cost base once in minus one. This is not Parliament minus Lululemon will be 0001 So when we simplify this further being cared for by his outside this will be vice were minus two means to since be by square minus force for the answer This group I buy square So this answer off party this struggle will bark me. Another axis of rotation has changed Texas a probation. Our is X is equal to fight. This is axis of rotation. So when we take you use the show matter the VDs and this case will be me. The distance would be this Onder since this values X on his compete values by so this particular value would be buying minus six. So being degraded this we get no volume Expressionists by artist by menace X and y's X sine x deitz in the limits Zero moment. Degrade this. We can't do by on over here. That's one of the brackets. So we have by X say nix DX minus you fight into your life extra square slang Knicks DX When the Lakers, for the bigger through by sweat into girl, affect sane X DX minus two play on. We already have been Diggle off Excess Square Sine X, which we just checked in the first part. So this is the excess stupid extra square. Say, nix the value of two basis quest. I need to stoop. I buy square minus force. We just replace a traitor. We just have to do the first part. So that would be in violation. My parts for stone taken down. So this will be Dubai Square the first time. That means that it doesn't develop Second of remains prospects. I simply love defense. Station of customer one develops minus. Call six months to get into greatness, Find a store by cube. Let's eat by the interest opened of the brackets. So this photos interface to a place where my ass sex bar stakes. Here you have a last name and the girl, of course, the same. And the people elements of zero reply and the stream means asked us. So when we put the parliament, we have minus by, caused by his minus ones. Was plus signed by zero minus. Listen, become zero Anderson become zero minus trip like you. Let's see. So this finally becomes by Q minus Group I Q. That's a place for the final answer. It's these two gets Castle

Okay, so our volume is this gonna equal to high in a row from 00 on a seven of x times our function with the B to the X B x. Okay, let's let you equal to X and you is equal to the ex. The DVD is equal to the e to the X, the X and the easy with tricky T rex. So we have in a row of x e x is equal to x e x minus. The integral of each of exodus is X e x minus east Rex. Okay, putting that back into what we have. We have two pi times. Our solution no evaluated f's or endpoints. Ellen of seven. And so that gives me two high times. Ellen of seven. You to the island of seven. What? Excuse me? Seven. So it's seven of here, minus seven in minus too high. Times zero minus each. Is there a which is one. You get 14. Hi, Ellen of seven minus 14 plus two fine lettuces. 16 pie owen of seven minus 14 as a

Okay. What we want to do is walk through the process of being able to find the volume of the region banded by Weichel the nine minus X squared. Um For X values greater than or equal to zero, X equals zero. Y equals zero about the Y axis. And any time we were talking about finding areas or volumes, it's always a good idea to draw the graph of the region. And so we're going to go ahead and draw that graph, it doesn't have to be perfect. Um We know it's going to be up here at nine. So if this is 369 and then it's going to cost the X axis that positive Three. So here is that region right here um without three and that is nine and we're revolving it about the Y axis. Um And so what we want to do is so it's going to be this region right here. X equals zero. Why called zero? And the graph um then the next thing we want to do is always draw that representative red rectangle. Right? So we have our little representative rectangle right here. Whoops. Uh huh. I can't draw a straight line for the life of me. Um So there we go. Um where this with is delta X. The height is of course nine minus X squared. And the distance from the axis of rotation is X. So um there is are a kind of a good quick sketch. So delta V is equal to two pi two pi X because that's the distance from or X is the distance from the axis of rotation. Um And then we are doing the height which is nine minus X squared. And then of course delta X. The width of that rectangle. So the volume is the integral from 0 to 3 of this. Let's go ahead and let me go ahead and distribute the experts. So this is two pi nine x minus execute. So this is going to be two pi times nine x minus x cubed and of course the delta X goes into dx OK, so now we can go ahead and integrate some a factor out that two pi and then this is going to be nine halfs X squared minus 1/4 X to the fourth and we're going to evaluate both of those at three and zero. Um So this is going to be two pi this will be 81/2 minus 81 over four. I'm in of course minus minus zero. Right, so everything is over zero. Um and so then this will be actually equal to, this is actually gonna be equal to um 81 pi over two and there is our volume.


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