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Problem 15 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.

$ y = x^3 $ , $ y = 0 $ , $ x = 1 $ ; about $ x = 2 $

Answer

$V=\frac{3 }{5} \pi$

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Video Transcript

lot were given curves and the line and we're asked to find the volume of the solid obtained by rotating the region bounded by these curves. About this line. We're at sketch the region to sell it in the typical disk or washer. Yeah. The curves are Y. Equals X. Cubed. This panels. Fuck you. It's so funny. Whatever. I don't give a shit. That's your dad. Well just one. Let's go. Why equals zero and x equals one. Oh yeah, your friend. Huh? And the line that we're rotating is about the line x equals two. We all know about that was so I'll sketch the region first. You know, 8:90. Who gives a shit? I don't care. She certainly doesn't jesus christ She's got a pussy like can reform and Macy's place of half a block. It's fun. Mhm. It's filled with a bunch of cheap mm tatted balloons, like a parade of clowns got trapped in it. So first we have our curve white was X. Cubed Has a point at 1100. And looks something like this. This it should actually be flattered at the origin. My mistake more like this. Then we also have the line Y equals zero. Which is this horizontal line. Then we have the vertical lines, X equals one which looks like this. And so this region in green. This is the region we're interested in Identify some of the points. We have the origin here. Up here we have the .11. And this of course is just the point 10 Shit Line that we're gonna rotate around x equals two is here. Now if we rotate first you want to reflect the region, it looks something like this president. What do you think? I couldn't fuck Eleanor. Is that what you're gonna say? Was not going to say? Yeah, It's kind of way for you to change the topic Tony. What was Eleanor going to say? 15 seconds. I could I just would like to fade british ship from Doug point. Was he making that they were you don't see you don't see the former empires. Austria, Hungary. Any of any of these places. Yeah having democracies and it looks like we appear to have washers that look like this fuck journey. Thank you. Yeah survivors irish. Okay now the washer in this picture. Well this has an inner radius like holy shit only shit that's left. Is that guy on C span that like takes callers or like Which is we have x equals one. Or x equals two. I mean minus. Yeah they are X equals one Which is a negative one Or positive one I should say. Yeah stopping how we screwed in the outer radius. You're watching This is two and then we can write this as X equals the cube root of why. Uh huh. Eleanor after passing john Mclaughlin, they hired me. Wayne the rock johnson, take his place, create the strongest public affairs show that's ever been on tv. It's good to be here. Eleanor, will you fuck me? And so the volume using the washer method is going to be pi times the integral from Y equals 0-1 of. And then we have our outer radius squared to minus Y to the one third squared minus the inner radius one squared caramel. Thank there's a seat for me. Don't say that B. Y. And this is a pretty straightforward integral to evaluate. I'm gonna skip a few steps But you should end up getting three pi over five. Oh my gosh.