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Problem 10 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.

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

Answer

$V=\frac{248 \pi}{45}$

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Video Transcript

were given curves and a line of rotation and rectifying the volume of a solid obtained by rotating the region bounded by these curves. About this line were asked to sketch the region the solid and a typical disk or washer. So we're given the curves X equals negative two minus Y squared And x equals y to the 4th and the line of rotation. Well this is about the Y axis. The street sucks dick. So the first curve is a parabola. This is a problem that opens to the left and has an X intercept of negative two. And then the other curve is a horizontal cortex essentially and it opens to the right. So it looks something a bit like this shit. Okay. No. Right. Yeah. Yeah. So our cortex looks something a bit like this but we did ask them. Yeah. And then it became wait to be sure. So I think there's a mistake in the way the the question was written on the website, was it? This shouldn't be X equals negative two minus y squared. It should be positive to minus y squared. This way there is a region of overlap between these two curves where there previously was none. So our region will look something a bit like this. Yeah, we're really just concerned about this region here in red. Yeah. Right. So there are a couple more points of interest on this region. You want to know for example what this top point is and this bottom point is. Well, they pretty clearly are the solutions to the intersection of these curves. So these are the solutions two to minus Y squared equals y to the fourth. Right? See what and a lot of like driving. I did. So if you solve this equation, well this is why do the fourth plus y squared minus two equals zero or y squared plus two times Y squared minus one equals zero. Or We just get y equals plus or -1 AIDS. And of course we plug in, y equals plus or -1. We get the x just equals one. And so we get the two points. 11 and 1 -1. These other two points. They are the origin 00 and the .20. Yeah. Most part. Okay fact that I could talk my way out on the wrong side. Now the line that we're rotating around the Y axis is this line in green the realize that and now I'm gonna rotate. We got pulled over like you turn on off red. Yeah for being said try to find P. F. Chang's. Yeah. And the cop was like uh like have a good day boys. Yeah, I remember that. The comedy club in Austin in the hotel that was wow. Mhm. But it was with the drought. Yes. Yes, yes because like that took a more extreme options there and the club is right next yes underneath. So the solid looks something like this. If you look closely, cross section of the solid is actually a washer like this. Mhm. Comic in green. Okay. Absolutely. To the hotel which is like you could walk like this A. U. Turn and while they sorry I'll redraw the washer and actually looks something like this and it's just like different a lot. First in the interior of the washer looks something like this. Yeah, it's kind of hard to see but there is an inner and outer radius. So we have a washer with an inner radius never. And this is why to the 4th and an outer radius which is two minus y squared. Therefore using the washer method are volume V. is equal to pi times the integral from why equals we go as low as negative one to as high as positive one of the outer radius, which is two minus y squared squared minus the inner radius Y to the fourth squared dy. This is just a polynomial expression. It's pretty straightforward to find this definite integral. If you do, you should find eventually, but this is equal to 248 pi over 45 as many games it's worth, yeah.