Problem 6 Easy Difficulty

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.

$ 2x = y^2 $ , $ x = 0 $ , $ y = 4 $ ; about the y-axis

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RG

Raymond G.

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it's were given curves and a specified line and rest to find the volume of the solid obtained by rotating the region bounded by these curves about this line. Also, we're asked to sketch the region the solid and a typical disk or washer. The curves are two. x equals y squared Parabola. As well as x equals zero. And why equals four. And the line we rotate is about the y axis. Just in the spirit thing. So first I'll sketch the region. Those hate balloons. Oh God. I think I'm gonna be sick. And so first I'll sketch the parabola. What's the more you this has its vertex at the origin and also has a point. Uh Yeah 22 just they didn't even have fast. Yeah. I mean the kid was, I'm not saying it's good as well as too negative too. Yeah. You're right. Would take murdered is worse than thing. It's also they were. Yeah, We have the vertical line. X equals zero. It looks like this and we have the horizontal line Y equals four. Yeah. Going to disgrace the thing by learning how to suck that good to actually save the hope and I would love to be a prisoner. It's just a strong and so this area and read this is our region. So actually delete this lower part. It wasn't really necessary after all. Where Now let's figure out some key points for this region. So we have the point here at the origin 00. We have this point here Which is 04. And then we had this other point here on the right we know what the why coordinate is it's four. Well this is two X equals four squared equals 16. So that X equals eight. And so this is 84 I know. So now we reflect about the Y axis, the Y axis. Is this vertical axis here. And to draw the solid. So al mir across this axis. Mhm. You have a plan? It's not. And then thank you, Take it back draw circles like this. Uh huh. This is sort of a sketch of what are solid looks like. It might be able to draw a little neater but a But you a typical disk looks like this in green. Take it out. Now we use the disk method so that our volume is going to be the integral from Y equals zero To y equals four of the area of the disk, which is pi times the radius which are radius. W. W. Well, we need to solve our equation for why. So we get Y equals the square root of two X. If we're just looking at positive why? And so this is going to be two pi. This is pi times route two X squared DX. No this is incorrect. Big fuck. They'll just do what boomers set up. Millennials. Millennials and boomers are exactly Oh yeah underneath the exact same. Yeah we're really doing it. We're really changed and then give okay sorry. Instead I should solve for X. X. Is equal to Why squared over two. So we have pi times Y squared over two squared. That is yeah. Just fuck revenue at full speed. Dy. Mhm. So this is equal to Pi over four times taking the anti derivative. This is 1/5 of why? The fifth From 0 to 4. This is the same as Pi Times 4 to the this Over four times five. This is the same as 256 pi over five things had