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.
$ xy = 1 $ , $ y = 0 $ , $ x = 1 $ , $ x = 2 $ ; about $ x = -1 $
$(2+2 \ln (2)) \pi \approx 10.638$
Applications of Integration
we are given curves and the line and rest by the volume of the solid obtained by rotating the region bounded by these curves about this line. Whereas to sketch the region the solid and a typical disk or washer from the solid. So the curves are X. Times y equals one. Y equals zero x. equals one And x equals two. And the line that we're rotating is about the line x equals negative one. It's so for can you sketch the region just to smell it late on? Just to have a little bit of piss to microwave and slip on while you beat off at home And then shipped. Glad fucking tupperware and in 30 in 30 seconds on your fucking preset on the way and then putting a towel over it. Got him. Yes. I think you gotta Eleanor, you looked wallet but I was ovulating you mind using the bathroom. I'm first of all I'll draw part of the curve. Xy equals one. We obviously have a point at 11 They're also the points at 21/2. Sand at 1/2 to the curve looks something a bit like this. Gonna Now we also have the line Y. Equals zero which is a horizontal line here at the origin. We have X equals one which is this vertical line and X equals two. Which is this other vertical line. And so this region and the red here. This is the region that we're interested in rotating. Ah I'll go back and watch a couple laps of the old. Let's identify some of the points on this region. So this top left corner is a .11. What This top right corner is to 1/2 And then these are of course 10 and 20. The line that we want to rotate around is the line x equals negative one. Which is here. Yeah. This jet like this shit is taking forever. Still waking up. I'll begin by reflecting the region across this axis. I went to bed. I tried it. Yeah. It looks something a bit like this. Maybe you should try something or did and then, hmm rotated solid. Look something like this. Maybe you should give us me and next heaven maybe you should suck my car. Adam friedland, suck my dick. But on the group told me tonight on the fucking group. Who gives a shit fucking bitch. Yeah. Uh huh. Whatever. Michael read the fucking car. Whatever. I don't even care. Give a shit. I think I care about that business. Yeah. Right. Give a fuck now. Looking at the solid, it's clear that we have a washer. Looks something a bit like this. Yeah. Don't ask her about it. Put your phone away, You fucking whore. See? And there's actually two different sections. So in this upper section we have a radius ray washer. Anything that looks a bit like this. Mhm. No, don't ask bitch. Don't. Well she's lying. We thought she's lied. Hey? She's a horned. She's embarrassed about it. I'm like you sorry. She's being former lady about it. And we have another washer down here in this second section. Yeah. This is he was very very total. No. I thought bye. I roll on mike. Yes. Sure. Yeah. Mhm bro. I hope you will die. We done. Yeah. So we really have two sections to integrate over many moms tomorrow night tomorrow. Funny mars maybe I should probably books for the top washer. We have an inner radius. Cool. What? Which is? This? Is X. equals one. My S. X. Equals negative one. Which is a radius of two. And the Albert radius right now for this is X equals one over Y minus X equals negative one which is one over Y plus one. Now for the bottom washer, Well this has an inner radius once again of two and the outer radius is now. So um instead of one over why this is X equals two minus negative one, which is three. That's the show. And therefore we have the volume Using the watcher method along the y axis. This is the volume of the top. What's the volume of the bottom? Which is high times the integral from Well, we're gonna integrate from y equals 1/2 to y equals one of our outer radius for the top which is one of the y plus one squared minus the inner radius two squared. Dy plus again by the Washington. And this is pi times the integral from y equals zero to y equals one half of our outer radius, three squared minus or in a radius just still two squared. Dy. How it started to slip there. An injury that I sustained while arguing with? Yes, this is pretty straightforward. It's remember anti derivative of one of Hawaii's natural log of Y after many steps and using log properties. Yes, this will simplify to two plus two natural log of two times pi and if you need to plug this into a calculator, this is approximately 10.638 notice I'm saving many steps here screaming.