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Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = x^3 $ , $ y = 1 $ , $ x = 2 $ ; about $ y = -3 $

$V=\frac{471 \pi}{14}$

Applications of Integration

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Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Boston College

we are given curves into line and he has to find the volume of the solid obtained by rotating the region bounded by these curves. About this line. Were asked to sketch the region the solid in a typical disk or washer. The curves are Y equals X cubed. Okay, Y equals one. Index equals two. Right? And the line we're rotating is about the line Y equals -3. No, you say. Yeah. Where were you? Yes. So, first we'll sketch the region crisis is No. So yes. So first I'll sketch the cubic function bombs to sorry, has appointed the origin the inflection point where it's approximately flat and then increases like this. Right? Yeah. And we have a horizontal line. Y equals one go. Yeah. About here. Oh, so you're a sub, Does it sound like it? The line x equals two. No. Well this is this vertical line, Right. Mhm. This is going to be off the graph, but it's a pretty steep. Mhm. Uh All across frost steve region. It looks something like this. Have any uh I've got for this region. We want to figure out what some of these points are. For example, this point here is 1 1. This point here will it has the X coordinate of two and appears to have a wide coordinated as of one as well. This uh point up here. Well, we know it has annexed coordinative too, but we don't really know exactly what the why coordinate is. But if you plug in X equals two, we get Y equals eight. Awesome. He's got all that make The line going to rotate about is why equals -3. Look at they like spray. Get sprayed. Right? So this is our horizontal line. Y equals negative three. We're going to rotate around. It is very far down. And the guy, it almost doesn't even make sense to draw this graph. But so be it. Yes. Now I'll draw the solid as best as I can and blue first I'll outline the region and then I'm gonna reflect the region across the axis. Mm That's already yeah, here in new york research actually. Yes. Wait yeah, no, these guys who are HIV. Let's see. So it looks kind of like this 6000 square Yes, 6000 sq ft. Yeah, that are you one of those guys? Yeah, yeah, that do you think about? That's Yeah, I mean mhm. Would you give me a hand sure after the show? Yeah. Yes, Judah gala and what? Yeah, yeah, yeah. Dude impressions maybe you and joe list on Cleveland like yeah, they do a character. That's okay. I'm not sure. I can see the name. Look joe. I love that. I love that boy. Oh you have like ah it's the last oh just a baby. This baby girl. Was he a so it looks something like this? Please call Wendy lamb my vagina. This is where I'm a baby. A baby with a baby's person. I'm afraid the graph isn't going to help illustrate the surface very well or the solid. I mean, but it should be clear enough that we can tell This is actually the baby adults person. Uh there are washers. I can draw them in. Green ma'am. It's a girl and she has an adult's pussy, ma'am. I'm afraid to inform you your baby. Within that prosecute getting this. It's Mhm. What? Mm. I think what's called adam. Well down Judah shoot us. Right. That's me. Were pressed. Mm. System. Okay. He's just fucking shitting. Doesn't even know Adam. No. He's just going to do in spots. Yes. The gayest thing you've ever seen this comment? Yes. They have almost the same shaking and your kids. We haven't. What about a show Highlander in the cloud. He's second guys off throughout history and sometimes never chuck. Right. Sure. So it's hard to see. But there is this green washer and so we see that the washer has an inner radius questions. Doesn't doesn't highlight. He always. Which is one minus negative three, which is four. Yeah. And we have an outer radius right? Which is our function. Why equals X cubed minus negative three which is x cubed plus three. Let's wait. And therefore by the washer method are volume V is equal to pi times the integral From what we see. X ranges from 1-2. Mhm. Of our outside radius X square sorry X cubed plus three squared minus our inner radius. Four squared dx. This is a pretty easy integral to evaluate. You should be able to do it on your own, and once you do, you should get an answer. Let's see the of 471 pie over 14 pancreatic cancer.

Ohio State University

Applications of Integration