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Chapter 6, Section 6.2, Question 003Incorrect _Find the volume of the solid that results when the shaded region is revolved about the indicated axisY=9 - 2xEnter th...

Question

Chapter 6, Section 6.2, Question 003Incorrect _Find the volume of the solid that results when the shaded region is revolved about the indicated axisY=9 - 2xEnter the exact answer as an improper fraction if necessary:2171 12EditClick if you would like to Show Work for this question: Qpen _Show Work

Chapter 6, Section 6.2, Question 003 Incorrect _ Find the volume of the solid that results when the shaded region is revolved about the indicated axis Y=9 - 2x Enter the exact answer as an improper fraction if necessary: 2171 12 Edit Click if you would like to Show Work for this question: Qpen _Show Work



Answers

Find the volume of the solid that results when the shaded region is revolved about the indicated axis. (check your book to see figure)

Trying to find the volume of this shaded region right here, doing this about the X axis, going to be left with a hole in the bottom there and so this is going to be the washer method. Okay. Which means that we're going to take High times the integral from 0 to 1. Uh huh. Two minus X squared squared by this X squared since the two minutes X squared is the function on top and the X. Is the function below here. So that's what we'll have the X. And then let's simplify that first. So we'll have before my ass or X squared plus Next to the 4th minus X squared. Now go ahead and take the integrator from here. So we get forex And so these we can combine. That's the -5. So negative five, execute over three Plus X to the 5th Degrees five. He plug in from 0 to 1. Bye bye. Afterwards You have 4 -5/3 plus 1/5 times pipe. So that will result in 38/15 pie.

Find the volume of this shaded region here, not around the indicated access, which is the y axis. So to do that, first of all we have to solve for X for this equation here. So we get Y -3 and then we divide by a negative too single tax, so therefore X is equal to negative Y Plus 3/2. So we're gonna take that in terms of the boundaries of the Y values, which is from 0 to 2 pile on the outside. And then we put in that equation because it's flat against the Y axis here, this is going to be the disk method which take just this squared. So let's go ahead and simplify that first. Sure to to buy. So taking a five plus three squared is y squared minus six, Y plus nine. All of that divided by four. Okay, so let's go ahead and take the anti derivative from here of pi times. That's why it. To the third, four times 3 is 12 and six. and 4 here Gives us 3/2. Yeah, and that's three I squared over four. And plus that's why. So that's also over four. We're going to take that from 0-2. Oh looking in and we get 8/12 minus 2, 4, 3, 12 or four. The speed teen over four. Spy. We simplify that fully. We kept 13 for six but

We want to find the volume of this she interviews right here, just marked in red and we're doing this cross the over the y axis here, that's the excessive irritation. And so with that in mind here, we're going to have this whole they're so therefore we're going to use the wash your method here. Okay? So the first thing we need to do is get this in terms of X. So that's going to be X is equal to one of her wife Because X can brought over sex. Michael's one. Therefore right next time, is that this Y value right here? So we plug in two for X. I get why is one half? So that's going to be one half right here. Everyone is going to go from 1/2 22 by on the outside here. All right. And so the We go from the highest x value here, which is the two. So that's where we're going to use to sit as our boundary. So it's going to be two squared. Okay? And then from there we subtract the one over Y squared. Right so now simplify their first It's gonna be 4 -1 over by squared. Do you what? Let's proceed to take the anti derivative of this next. So that's going to be able to pi times for why? And then an iterative negative one Y squared becomes uh -1 over why? But we have a negative ready switch turns positive. Mhm. It's like this do that from would have to to so plugging that in we get eight plus one half -2 -2. Right? Plugging in the one half their flips it that's how we get the negative two at the bottom there on this one. So then So it's going to be equal to 4.5 pie Or 9/2 pi.

Trying to find the volume of this region. two equals 0. So the sketch for this. Yeah. Here. And we're planning this shaded region Which actually goes from -32, 3. Do you want to point out that from 0-3 is actually going to be the same area from Date of 3: zero. So if He contends that that would be from 0 to 3 two times hi of Then that's going to be 9 -6 squared squared. Right? We're going to be using the disk method here because it's just straight revolving around the X axis here. There's no gaps between the function and that. So, okay, Taking into account here, we've got two pi through 23 simplifying That. Before we take the integral. So it's 81 -18 x squared sex fourth. Let's take the anti derivative of this two pi from 0 to 3. Yeah, two pi times 81 X. Lightness. That'll be six X to the third. Because we divide by three 18, # three is 6. We have plus X to the 5th over five charge to three. So then we get to 43 minus 6 27 first 3 to the first floor of five that tops to pipe. And what we would get Is 1296 precise. That's fine.


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