find-the-volume-of-the-solid-obtained-by-rotating-the-region-bounded-by-the-given-curves-about-th-30

Question

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.
$$y=x, y=\sqrt{x} ; 	ext { about } x=2$$

Rebecca Polasek · Accepted Answer

we&#x27;re gonna find the value bounded by these graphs that I&#x27;ve already sketched out. And first figure that out. We want to look at the area. So the area between these equations is this part of shading and green, and we want to find volume going around X equals two. So this is X is one this is excess to and remember X equals as a vertical line. So if I&#x27;m finding the volume rotating around X equals two notice it&#x27;s gonna be a washer set up because there&#x27;s all this missing space as you think about it in three dimensions, you&#x27;re getting kind of this doughnut bagel sort of shape as it wraps around with this big missing part in the middle. So that&#x27;s a washer set up. So for the washer set up, you know, volume is pi times the integral of R squared minus R squared. And as you&#x27;re thinking about your variable, you want to think about really what access you&#x27;re working with An X equals two is like working with the Y axis. Notice how those are parallel. So I want to use d Y, which means my limits here within a girl are also in terms of why to the first name on the label is the Y values that bound this area. Well, the lowest Y value, I hope is obvious, is why equals zero. Because these two graphs meet at the origin. Thes graphs also meet at the 20.11 So the final y value here is one. So for my inner growing up I and then integral from 0 to 1. So then the next thing I need to figure out is what is capital R and lower case R for capital are I want to think about what would have generated a volume that had no missing pieces, a solid volume. Well, that would be if I went all the way from the edge to that axis of rotation the whole way up and down. So if I color it in like this, see how that would be a solid even as I spun it around Well, thinking about kind of like you did with area right minus left for this horizontal set up, that shaded in portion would be the vertical line to minus the curved, which is the square root of X. But remember this in and girls in terms of why So I don&#x27;t want to write the square root of X here. I want to solve this to be in terms of why, in other words, make it an X equals equation. So you square both sides and realize that curved graph is really written as why squared in this case. So for capital are I&#x27;m gonna put two minus y squared. It&#x27;s the same function. It&#x27;s that square root equation. But I wanted to write it in terms of why so having to solve for X equals to get that written as Why squared? So for lower case, are we think about what would we have to take away? Well, this stuff I&#x27;d have to take away would go from two. Sorry, that&#x27;s still little are goes from two to the straight line. The linear function is acts well again. I don&#x27;t want to write this in terms of X, because my inner girls, in terms of why thankfully, here X equals wise. There&#x27;s not a whole lot of to do to make that switch. I just need to write it as why. Because why was ex So this is my set up to minus y squared that quantity squared minus two minus y squared. If I were to go through and it&#x27;s all this by hand, I would need to expand thes by no meals so you would want as your squaring them. You&#x27;re multiplying them through your distributing all those terms. But if you have the option to calculate, this is nice one to be able type in. And this integral portion becomes 8/15 so that if I multiply for the volume, I would write eight pi over 15 and that&#x27;s my final answer.

Rebecca Polasek · Answer

So I sketched out to set up here for finding volume, and we&#x27;re going to rotate this volume around. Why Ankles, too? So the equations were given. Why equals one X equals two and then why equals e to the negative X And that&#x27;s just a decay function there. So first I want to label the area that were rotating the area is whatever is bounded by all three of those equations. So here, that&#x27;s going to be the participated in red. We want to rotate around the line. Why equals to? So let&#x27;s say that&#x27;s here. So we&#x27;re taking this whole part that&#x27;s in red and rotating it up and around. Why equals two. So it should be pretty clear that there&#x27;s all of this missing space As I&#x27;m creating that volume. It&#x27;s not gonna be totally solid. There&#x27;s gonna be like a doughnut. There&#x27;s gonna be that empty part in the middle. So because there&#x27;s that missing space, then we know this set up here is a washer set up. So your formula for the washer set up is high times the integral R squared minus R squared. You pick your very well. Based on what access you&#x27;re working with and here, like was too is like working with the x axis. See how those are parallel. So I picked D X for my variable, which means for the limits I need to use excess as well X one and X two. So now I need to go back and figure out Well, what is that? X one and x two? What is the starting x value in the final X value for the area? Well, if I&#x27;m looking at this area, the tribute just something I can inspect and see those X values as zero. And to write the given vertical line and the access says you were into then the next thing to figure out is what is capital R and lower case R Capital R A&#x27;s What if this volume wasn&#x27;t missing anything at all? So if I&#x27;m going from, why equals two to the shaded area and I wanted to say, Hey, I&#x27;m not missing anything, I&#x27;d have to color it in like I just did in blue, right? I&#x27;d have to fill in all that missing stuff, which means it would go from that horizontal line of two down to the curved line which is e to the negative x so thinking about it. But we do area upper minus lower here, I think, to minus either the native X for a lower case. R it&#x27;s what am I taking away? Well, the stuff I&#x27;m taking away is this missing block between our area and the line of rotation. So that comes from that upper bound of to minus the horizontal line one. So two minus one. Or you could just say one is lower case are so filling it in capital R squared to minus E to the negative X squared and then lower case R is wine. So one squared, which is still just one. And then at this point, we could expand that binomial. Maybe you have software you can solve this integral. So solving through. We get pi times for e to the negative too, minus e to the negative for over two. Plus I&#x27;ve over two and that&#x27;s an acceptable final answer for this volume.

Rebecca Polasek · Answer

so I&#x27;m gonna find the volume. If I&#x27;m rotating around, why equals one? And I&#x27;ve already sketched out the graph for y equals X and then the graph for why I equals the square root of X. So that means the area bounded by. These is this part here. And if we want to take it around the line y equals one. They were thinking about this horizontal line y equals one that were rotating around retaining towards. So because I see that there&#x27;s this big, empty space that I&#x27;m not using here, that means I&#x27;m considering this A washer method for the volume and for that washer method we know volume is pi times the integral of capital R squared minus lower case R squared. Now it is still with respect to X, the d X. I use here because why equals one is moving like it&#x27;s the X axis. So that means I&#x27;m using X is my variable. So now I need to decide where did the X values happen? What is the first in the last x for the area and then set up capital R and lower case R. So first I like to look at the X values. And if you know a little bit about these graphs, then you should know hopefully where X and square root of X intersect. If not, you can solve for where these two things intersect. But it happens where X equals your own X equals one. So I know my limits for the integral are zero and one now for the capital are you&#x27;re trying to think about? What would this volume be if I didn&#x27;t have any missing pieces? Well, I didn&#x27;t have anything missing at all. This volume would go all the way up to that line. Y equals one. See, high filled in with no space now well, that would be defined. Upper minus lower as one minus the straight line and straight line is X so capital are gonna call one minus X for lower case R. You&#x27;re thinking about what am I trying to take away? So all this part here in between that I just colored in is what I want to take away. Well, that goes from one down to the square root graph. So one minus the square Durex. So these are the two things I want to fill in for Capital R and for lower case R on this problem. If I were to go through and then fill it in my hand, I would need to expand thes by no meals. Remember, since its squared your actual think of it like it&#x27;s multiplying times itself. That would take a little time to multiply out. If you have the option to use a calculator, that&#x27;s definitely the fastest choice here. If I type this whole thing in for my inner girl, I get 1/6. So my final answer is pi over six for the volume.

Chris Trentman · Answer

we are given curves and a line and rest. Find the volume of the solid obtained by rotating this region bounded by these curves. About this line. Were asked to sketch the region the solid and a typical disk or washer. The curves are Y equals X squared. A parabola opening up X. Equals Y squared parabola opening to the right. And the line is the line Y equals one that we&#x27;re rotating around 1st. I&#x27;ll sketch the region. You say that. Yeah it would have been really woman. No. Holy shit with rock while you&#x27;re slicing fucking sword just coming off and then you take your sword breaks. You take away the curved sword. She just now we start off by drawing are Upward facing Parabola has a vertex at the origin and a .11. Then we draw The right word facing Parabola. It has appointed the origin or Vertex and also a .11. And so this region in red. This is our region poison. Now this one point is clear the origin. This other point is the .1, 1. The line that we&#x27;re going to rotate around is the line y equals one. This line in green. So if we rotate, you know like, well I&#x27;m going to mirror this across the line of rotation. Look something a bit like this. Who sometimes we&#x27;re saying romans one of the big things. Knowledge, Yeah, yeah, What else? Yeah, of the these hills. And first that&#x27;s why that&#x27;s why romans build roads is the angus Kononov. So they didn&#x27;t have little dad, they were good at. Yeah, there were horses. So this is what our solid looks like in blue. And we actually see that this is a has washer cross sections that look like this. His how far west today they were like Austria. Mhm. What? And we see that the washer has an inner radius. I learned. Sure, that&#x27;s right. With alec baldwin, that&#x27;s about genghis khan. Well, this is going to be one minus uh mm. What one of them is the only does that mean? No, for real Y equals square root of X. This is one minus root X. And the outer radius. But isn&#x27;t he is one minus X squared everything. Mhm. It&#x27;s his G. And therefore using the washer method are volume is equal to excuse me, pi times the integral from Well x ranges from zero up to one of our outer radius which is one minus x squared squared minus the inner radius, which is one minus the square root of x squared dx. This is a pretty straightforward integral, integrating square. It is the same as integrating polynomial. And so this eventually, after several steps should simplify to 11 pi over 30. Mhm. How do you think?

Chris Trentman · Answer

lot were given curves and the line and we&#x27;re asked to find the volume of the solid obtained by rotating the region bounded by these curves. About this line. We&#x27;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&#x27;s so funny. Whatever. I don&#x27;t give a shit. That&#x27;s your dad. Well just one. Let&#x27;s go. Why equals zero and x equals one. Oh yeah, your friend. Huh? And the line that we&#x27;re rotating is about the line x equals two. We all know about that was so I&#x27;ll sketch the region first. You know, 8:90. Who gives a shit? I don&#x27;t care. She certainly doesn&#x27;t jesus christ She&#x27;s got a pussy like can reform and Macy&#x27;s place of half a block. It&#x27;s fun. Mhm. It&#x27;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&#x27;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&#x27;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&#x27;t fuck Eleanor. Is that what you&#x27;re gonna say? Was not going to say? Yeah, It&#x27;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&#x27;t see you don&#x27;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&#x27;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&#x27;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&#x27;s ever been on tv. It&#x27;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&#x27;s a seat for me. Don&#x27;t say that B. Y. And this is a pretty straightforward integral to evaluate. I&#x27;m gonna skip a few steps But you should end up getting three pi over five. Oh my gosh.

Rebecca Polasek · Answer

I want to find the ball. You given the equations y squared equals X X equals two. Why? And I&#x27;m rotating around the Why axis. So for both of these graphs, air really probably not in a typical form, we would see for graphing. So let&#x27;s talk about each of these shapes. First of all, why Squared equals X is, ah, horizontal crapola, And if you recognize that you know it&#x27;s the problem. Form something squared and it&#x27;s horizontal and you know that a little bit, and you could just go straight into it and draw your horizontal problem. But if you weren&#x27;t sure you could actually solve this for y equals to get y by itself, you would square were both sides. But remember, when you take the square root, you need plus and minus, which is what gives us the upper and lower parts of our problem. So the square root graft, I hope we know, is this upper part, and then the negative of it is what makes the lower half of the problem. So both options here for graphene. Good choices. Now, for the 2nd 1 I would just do a real quick rewrite here in solving for y equals because then it&#x27;s just really obvious that it&#x27;s a line that goes through the origin because there&#x27;s no plus remind of something. And it&#x27;s got a 1/2 and from the X. So it&#x27;s a slope of 1/2. So line through the origin slope of 1/2 exactly gonna start less steep than the proble and then eventually cross through it. So that would be my linear function there. Which means the area where considering is this area bounded between the two crafts. We&#x27;re revolving it around the Y access. So there is definitely, without a doubt some parts here that are missing. That means we&#x27;re using a washer. So because I&#x27;m easy in the washer set up, I use volume is pi times and a girl Capital R squared minus lower case R squared. We still use whatever variable where we are revolving around. So because it&#x27;s the why access I used d y. And that means my limits should also be in terms of why, like, why one and why to to the so the first thing we need to consider is what are those limits? What are the Y values that bound this area so I would take my equations for why? Why squared? And two White and I would let those equal to each other solving this. I would subtract the to y and then factor So my two solutions would be Why equals zero and y equals two. And that looks pretty good considering my graph looks like a cross around the origin. It was my hope with the problem, the straight line and then sure, this could be two steps up. Just depends on how zoomed in we are. So my integral going from 0 to 2. Those are the why values that we want to use to fill in. Then the only other thing I need is capital R squared and lower case R squared. I do feel like it&#x27;s easier to find lower case. Our first. It&#x27;s the stuff we take away. So it&#x27;s wherever we go from our access to still that empty space. So notice here, this is lower case are this is the stuff we&#x27;re taking away. And if I go from right to left, this is actually the parable a shape and this is the access. So lower case R should be that parabola shaped to the axis. Now, how do I know I want to use? Why squared and not the square root of X? Well, easy. If I filled in the square root of X over here, I would have the wrong variable. So I know I need to use an original form of the equation. Which is why square now for capital are we want to consider? What if this were a disc like, What would it take to make this solid? So to go from our access Ah, the way to the edge of the shape. Well, to go from the access to the edge of the shape here, we would need to go all the way over to the straight line. So for capital are if I consider right minus left, then I want to go from the straight line, which was two. Why? To the access, which is zero again. I used to I and not the salt form of it, because I need the variable to be a why. And I realized I said those right, but I put those in the wrong places. So capital are is that to why and then lower case R is that Y squared? Hello. I have to match those. Okay, so now we&#x27;re at a point. Where would be pretty easy to work this out by hand for the integral If I just want to square these terms. So squaring the first term makes four y squared square in the second term makes wide of the fourth. And then obviously doing that anti derivative of my hand Not too bad. 4/3 y cubed minus 1/5 y to the fifth from 0 to 2, all that multiplied by high in front. And if we use the upper bound minus the lower bound here and we calculate we get 64 high over 15 and that&#x27;s your final answer.

Problem

The region enclosed by the given curves is rotate…

Problem 12 Medium 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.
$$y=x, y=\sqrt{x} ; \text { about } x=2$$

Answer

$$
V=\frac{8 \pi}{15}
$$

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Integration Review - Intro

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$$V=\frac{8 \pi}{15}$$

Essential Calculus Early Transcendentals

Grace H.

Anna Marie V.

Heather Z.

Kayleah T.

Integration Review - Intro

Definite vs Indefinite

James StewartEssential Calculus Early Transcendentals

Grace H.

Anna Marie V.

Heather Z.

Kayleah T.

$$
V=\frac{8 \pi}{15}
$$

James Stewart

Essential Calculus Early Transcendentals