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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.

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

$\frac{10}{3} \sqrt{2} \pi$

Applications of Integration

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Campbell University

Oregon State University

Harvey Mudd College

Idaho State University

insist there's like, yeah, we are given curves and the line and rest 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 of solid. In a typical dishwasher. The curves are X equals Y squared, X equals one minus y squared. That's why the KKK dresses up like ghosts. And the minute we rotate about is X equals through. 1st I'll sketch the region juice or we don't look at this show up. Okay. Sorry, I didn't know that this prime that we floated around it. Yeah, because no one we'll start really good while Sam's got some stuff appreciated. I want them to listen. Mhm. It's a So first I'll sketch X equals Y squared. This is a horizontal parabola with the vertex of the origin and the point at 11 and negative one negative one. It looks like this was like and then we have the curved X equals one minus y squared. This is a horizontal parabola with the vertex at 10 And opens to the left like this. It also has points at zero plus or -1 looks like this. And so the area that interested in the region is this region here. Greg. Yeah. What am I actually doing? So you have a come podcast. So let's identify some of the points here. For example to find the top and bottom points we want to find. Mhm. From these are the solution cities, intersection of these curves. This is a solution to the system of equations which is why squared equals one minus Y squared. So we get that why I swear it equals 1/2. And therefore why is it with a plus or minus? Uh huh. one of her too. The timer's up for that. So we're not talking about stop and the value of X is y squared. So we have the X. Is equal to plus or minus one over root two squared, which is just one half. And so these points are positive, one half, one of the route two and negative one of the two positive. Sorry, one half negative. One of the routes to It's so here's the line that we're going to rotate around to rotate this. I'll actually reflect the region across this line. Yes. Right. So that's something like this I watched last night, Bad news on netflix. It's alive where it's like john Malkovich is in it, john mackey three seconds at the beginning and the solid looks something like this. Then he turns a picture. It's about something like uh huh Rugby team crashes in the Yeah, it's bad. It seems like it's a me because it's a guy, we've got to start eating the other guys like are you freaking nuts? So have you, have you lost your gosh mind for here? You know that they don't No one has any uh no one else famous. Let's just get to the next work. Okay, he's family. This is what the solid sort of looks like. You can imagine. And there are washers for this salad. Hell yeah, bitch. Don't try to be stopped. That's good. Thanks. Hell yeah, bitch. He says that a lot, right? The one of the washers for example, looks like this. We should I should just go back to old episode package all of this scandal after like they do with shows now, don't It's well, yeah, yeah. People like to point, No, they don't know. They might be a lot. Yeah. Yeah. Probably actually have a studio audience and looking at these washers or what they see they have in their radius is Carmichael. No, that Salon. No, these are, the radius is uh, X equals three minus um, one minus Y suede, which is two plus Y squared. No lodge to tell. And the outer radius is X equals three minus a people's Y scrolling. He's really smart and therefore you can wash. Your method are volume is equal to pi times. And the girl from, well, why equals -1 of the route to to positive one of the route to of our outer radius three minus wise food squared minus the inner radius two plus Y squared squared. Sure, D Y. This is a pretty straightforward and they're brought to evaluate. I'm gonna go ahead and let you do it on your own after skipping several steps here. Final answer should be yeah, 10 pie times the square root of 2/3, which is approximately 14.81. It is Puerto Rican guy that steals bicycles and

Ohio State University

Applications of Integration