the-volume-of-a-torus-the-disk-x2y2-leq-a2-is-revolved-about-the-line-xbba-to-generate-a-solid-sha-3

Name: the volume of a torus the disk x2y2 leq a2 is revolved about the line xbba to generate a solid sha 3
Uploaded: 2019-10-21
Duration: 7 min 14 s
Channel: Numerade
Description: The volume of a torus The disk $x^{2}+y^{2} \leq a^{2}$ is revolved about the line $x=b(b>a)$ to generate a solid shaped like a doughnut and called a torus. Find its volume. (Hint: $\int_{-a}^{a} \sqrt{a^{2}-y^{2}} d y=$ $\pi a^{2} / 2,$ since it is the area of a semicircle of radius a.)

Question

The volume of a torus The disk $x^{2}+y^{2} \leq a^{2}$ is revolved about the line $x=b(b>a)$ to generate a solid shaped like a doughnut and called a torus. Find its volume. (Hint: $\int_{-a}^{a} \sqrt{a^{2}-y^{2}} d y=$ $\pi a^{2} / 2,$ since it is the area of a semicircle of radius a.)

Bobby Barnes · Accepted Answer

So the disc expert plus y squared lesson to a square is revolved about the line X is equal to be or be a strictly larger than A, and doing that will create a solid shape like doughnut, which we call a tourists. And we want to find the volume of this and were given the hint that the integral from negative eight A of the square root of a squared, minus y squared divide is equal to pi times a squared over two. Since it&#x27;s just the area of a semicircle with radius, eh? So that hint is probably a good place to start. Um, first, what we need to notice is that we are revolving around this vertical line. And so in this chapter, anytime we wanted to find the volume rotating about Burke Alliance, What we want to do is used on the high are, um I won&#x27;t use a in this case, are you see, um, some outer radius which we call our and we want this in respect. Why? And we swear that. And then we want some inter radius which we call little our prospective wise wo square the And we interviewed this with respect to you. Why? So first let&#x27;s re bite this function in terms of or in terms of, Why? So that means we could be right. This here as X is equal to plus or minus a squared minus y squared, square rooted. And remember, this here would be, though positive side and on this side would be negative side. And we actually already know just from knowing what the shape of this crap is, what our balance of integration should be, since we&#x27;re ranging from values of age today. So just go ahead and write that in real quick. And if you weren&#x27;t sure that we could just look at the rain or or the domain of Oh, so you have the range of these, please. So it should be negative color, negative A to a so first step, but we want to do is figure out what our inner and outer rady I are going to be. So if we start from this moon line here, we&#x27;re going to go until we hit our first her, which is going to beat me the positive portion of this so we&#x27;ll have a little r of why should be so since V is on the right, we&#x27;re gonna put me there and then we&#x27;re gonna subtract off the positive of that. So negative a squared like that, then we&#x27;re going to have a capital R. Why is equal to so I&#x27;m gonna do the same thing. But we&#x27;re going to go until we hit our second curve. So just started here. We&#x27;re gonna go straight through that, which is just the negative of the square root. So once again, we&#x27;re starting from begin and then we&#x27;re going to subtract all the negative of that a square minus y squared. But these negatives here cancel each other out. So we would just have B plus square root of a squared minus wife&#x27;s would. And now, since we know all of that, we can go ahead and plug in our inner and outer radius into this So high negative a two way. So first we have a B plus the square root of a squared minus y squared, squared and then we&#x27;re going to the truck. Off are inter radius, which was the minus. This we&#x27;re route of These were honest, twice weird. And all this needs to be swear. And we integrate with respect to why. And now I don&#x27;t want to bore you all with the algebra here, so already did it earlier. But it should turn out to the pie. Negative A to a of for being square root of a squared minus y square. Why now? Let&#x27;s go ahead and pull out that four beat So we get or be pie for the pie Negative A to a of the square root of a squared minus y squared do you want? And now we were told that this here from our hint should be pie a square over too. And that&#x27;s just from our hint. So we get four b. Hi, that was Well, hi. A square over to And then we can simplify that too. Pie a squared B and I still need a two right there. So the volume of this tourists only rotate around. The access acceptable to be is to pi a squared

Patrick Vaughn · Answer

in this problem. We want to find the volume of the tourists obtained by rotating the Circle X squared plus y squared equals pay squared. It&#x27;s so here is my circle. It&#x27;s very good. And we&#x27;re rotating it around the line. X equals B. So here&#x27;s be rotating it around. So we get a doughnut much way we are, given that a ISS so zero is less than a is less than B strictly So, first of all, we what we could do to make this easier on ourselves as we can just integrate the top half of the region rotated around, then multiplied by two at the end. Well throughout the process. So we have V is equal to two times the inter role from so X is going from negative A to A because that is the radius of our circle. We have two pi Now what is the radius of our, um what&#x27;s the radius of our so wonderful shell? So when sure, So when X is negative A. We&#x27;re gonna have B plus a. And when X is a we have B minus a. So then our radius is gonna be be minus X. So next What is the height? So the height is gonna be the square root of a squared minus piece. A squared, minus export DX. It&#x27;s okay. So then this becomes equal to two integral negative A to a of two pi, and we&#x27;re going to distribute this. So we get be squirt of a squared minus X squared minus. So let&#x27;s pull it to pie out front. About four pi. So be route. Hey, squared minus x squared. Um, plus sorry. Minus x route A squared minus X squared DX. So in the first term way need to use a trick substitution. And in the second term, we can use a U substitution. So for here, right, we&#x27;re gonna want X is equal Thio. So X is going to be equal to a sign of data on D X is equal to a co science data D data. So let&#x27;s split these intervals up. So we&#x27;re gonna have four pi b times. Negative A to a of. And when we do this substitution. So we&#x27;re gonna get squared of a squared minus a squared sine theta something. Science court data. Sorry. Um, times a coastline data data. And so in this one I&#x27;m gonna you is equal to hey squared minus x squared. So then d u is equal to yeah, right and Oh, yeah, We need to look at the ones here. This isn&#x27;t that so. There should be, uh, naked. So a sign of theta equals negative A When sign of data is gonna be so negative. Pi over two tau pi over two. So that will be our limits. Because that negative pi over two, we plug that into a sign of data. We would get negative a a. And when we plug in pi over two to sign a sign data, we would get a Okay, so it faces that substitution. So here we have a squaring my sex words. So then d u is equal to negative two x dx And what does women&#x27;s go from? So when we plug in, um, negative A for X we get you is equal to zero. And when we plug in a for you, we get zero. So that goes from 0 to 0. Eso this term goes off to zero e shot right there. So we should look at that has over. That works because it&#x27;s non function because we have this X in front. Okay, so this term is going to zero. Um, next. So we&#x27;re tryingto find that so we can pull in a out. So I have one minus sine squared data. That&#x27;s equal. Thio Cosine squared data. So then this is gonna be four pi b A squared, integral from negative pi over two tau pi over two, uh, cosign square data D data. So now we need to use a trick identity to reduce this down. Thio. So we get four pi b a squared go from negative pi over two Thio pi over two. So that&#x27;s gonna be, uh, one half plus one half co sign of tooth data D theta. Right, So we get four pi b a squared times one half data. Oh, minus 1/4. Sorry. Plus 1/4 of sine two theta evaluated from negative pi over to two pi Over two, we get four pi. Let&#x27;s go ahead and distribute before so we have pie. Uh, times be a squared times two times pie because if we take pi over two minus pi over minus negative pi over two, we get pie. Um, plus and so the upper limit is going to be, uh, sign A pie, which is zero in the lower limit is a sign of negative pie, which is zero. So both of those go both of sign terms. Go away. Okay, so we get to pi squared B a squared, and that completes the problem.

J Hardin · Answer

a tourist is generated by rotating the circle X squared plus y minus capital are square is small, are square So that&#x27;s this blue circle up here and conference one into we&#x27;LL rotate that about the X axis And down here we have another circle and the volume will weaken Take a cross section here and because of this large gap in the middle, we could see that the cross sections or washers And we we could obtain this volume by letting the washer trace out this volume in the ex direction where since the circle has really is our we have negative are over here the X value are over here. So these are our bounds of integration, Rex. And now for the washer We know that we&#x27;LL want usually the general formula for washers, pie and then large radius All due notice by are big square minus are small So this is it for using d X. So these Andy or the X pounds of integration Now we&#x27;LL have to go ahead and find big radius and small radius. So looking at the figure here we see that we could the radius Well, here&#x27;s the center of the washer. Here&#x27;s the origin in and to find the Big Radius it looks like we go up here, Toe are and then we have to add in that semi circle. So here&#x27;s one way to find are big So let&#x27;s look at this and my circle, but placed at the origin so that will be explored. Plus, y square equals R square And then we&#x27;LL go ahead and shift up the circle by our units. So after we do that, we shifted up by Are the circle looks like this. So in that case, we see that the total distance it&#x27;s just are added to this upper half of the semi circle which is why equals from this equation right here from the circle R squared minus explode So are big We&#x27;ll just be are the shift plus the upper half of the circle it&#x27;s being radius and then similarly for the small radius are small Same idea but we&#x27;LL look at the lower half of the circle So this time we&#x27;Ll take that lower half circle and then we&#x27;LL shift it up I r and this will give us the smaller radius over here and dotage That&#x27;s a small radius. So this time since we&#x27;re on the negative circle our small B the shift up that we added in the vertical direction that&#x27;s our And in this time it&#x27;s plus. But we&#x27;re doing the negative part of the circles of this time. It&#x27;s negative R squared, minus explorers. So this is the equation for this Lower half someone circle exes. So this equation was the negative are slurred minus X squared. So we have found our big in our small, So we&#x27;LL go ahead and plug those into our volume formula. Let&#x27;s go to the next page for that. We have v picking up where we left off and we observed that the bonds were negative. Arts are in the ex direction. We have pie. Let&#x27;s pull the pie in the front of the girls. Before then, we have our Dade squared, so we have our plus the radical square the whole thing. So our big square and now we&#x27;ll do our small squares. Then let&#x27;s just go ahead and evaluate the squares so we have our square to our times the radical plus and then we square the radical. So this is the first square and then for the second square and then plus R squared minus x square. And then let&#x27;s go ahead and cancel out as much as we can. So we have our square minus R squared. Those cancel r squared minus x square and then with a minus part squared minus X for years. And here the radicals to two are radicals Won&#x27;t cancel because the double negative. So we get for our something, just go ahead and pull. The four are the constant for big Our pie. Yeah, in a rural negative. RTR and then we&#x27;re left with Where was that? Excuse me, R squared minus little exclude yet I sees me. I pulled out. The four are here and I&#x27;m left with r squared minus x word. And now, with ready for a tricks up. So here you should take extra PR sign. What? Thandie Exes are co signed data. Now here we have to be a little careful. We are dealing with the definite general. So these bounds these limits of integration are subject to change. So to change these, let&#x27;s go to our tricks up and recall anytime you sign entrance up years your restriction for data. So let&#x27;s plug in these X values. So the lower bound X is negative arm and then the right hand side is our assigned data. This means signed data is negative one and in December hold on ly time that happens is when data&#x27;s negative Piper too similarly for the upper bound X equals our plug. It in the left hand side becomes our and the right hand side. Our scientist, This means scientist is one and the only time that happens in our interval is that pirate too. So these air the fate of bounds of integration And let&#x27;s go ahead and simplify this radical before we go on R squared minus x squared. This is equal to the radical R squared minus our square science where cool it are. A lot are square and then we have a radical still one minus sign square. This is our times a squirt of coastline square and that&#x27;s just our co signed data so that the radical becomes our coastline data. But also DX gives you another Arcosanti data. So after multiply these arcos on Data&#x27;s together so that&#x27;LL be our next date. Let&#x27;s go to the next page we have for our pie negative, however, to two pirates who and then we&#x27;LL supplying the r co signed and our co sign We have r squared co sign squared data data. Now this little ours is not a variable That data is the variable so we can go ahead and pull the r squared out. And also for this coastline, let&#x27;s use the half angle identity Coastline square is one plus coastline to data all over too. So pull out the two from the denominator So you have four over to which is too big our pie little r squared and then one plus co sign to data we can integrate this integral of one Just data in a rolls co signed today No scientific data over to and we have our bounds negative power, Hutu power Hutu And I will just put those in Data&#x27;s pirate too. And then we&#x27;LL have sign of pie over Zoom And now we plug in negative however too plus sign of negative pie. Oliver too. We no sign of pie inside of minus fireball zero you have to capital are pie little r squared. Then you have pie or two plus a pie or two. So that&#x27;s just another pie in there. So you&#x27;LL have it times pi And then we could simplify this by just squaring that by multiplying goes together and there&#x27;s our final answer.

Vikash Ranjan · Answer

ex body was bilge Leslie. No, we know these. That these sip that is the shackle. Um, symmetry about the line exit was being blessed Lee. Therefore, we may conclude that the what is on two holding it off the center of mass is expanding. Plus be pleasantly. They&#x27;re full. They have also do. Then the idea. Yeah, used the formula. What? For? The area off a circle. Wait on equals. Therefore, any prints get by a Squire which is equal to by a Squire. Therefore by using park was too We have He was great. And is he going to Dubai B plus? Eh? In the buy is clan. Is he going to Dubai squired B plus, eh? India&#x27;s quite a city could do to buy his file A scar on the record aimlessly be. He said the answer to the given problem

Brian Ketelobeter · Answer

Good day, ladies and gentlemen. Um, they were looking a problem number 50 from section 7.3. And what it asked us just to find the volume of the tourists induced by, ah, circle of little of radius. Little are. And, um, the axes of revolution is around. Exp equals two. Big r. Okay, So let&#x27;s ah, let&#x27;s take a look at this now, The first. The first thing I usually suggest just to try and draw the diagram on and to get a good idea of what you&#x27;re looking at before you do anything here. So, um, in particular than if I if I draw my actually is here, um, this is my accedes. Except he&#x27;s in y axes. Um, and then my my actually is a revolution here. It&#x27;s given, um, around X equals big. Are So this is big are, um and then I&#x27;m Then I&#x27;m given that I have some sort of circle like this. Um, circle. Looks like it&#x27;s, um of course, this is negative. Are, And this is positive. Are, um okay, So now, uh, now there&#x27;s really two ways you could do that. Of course, you could use the shell or washer method. Um, now, this shell message, um, or the washer mess that I guess, Ah, would the instrument itself ago perpendicular to the axes of revolution. So in particular than you have the increment going like this, Um, you have the interment going from like this or the, uh I guess t shell method itself. Um, the rubber, the increment would go this way. Um, and ah, I guess it&#x27;s a question, really of preference. Uh, maybe now that I&#x27;m looking at it actually might make sense to do it the washer method. But, um, I&#x27;m gonna use the d. Ah, the shell methods. So, um, nowthe shell method. Uh, of course, we need the, uh we need the cops. We need the h of acts. So first off, the question is, what is h of X here? And when we we look at this H of X, of course, is tthe e the length of the increment here. So we look at the increment. It&#x27;s the length of the sky. Um and length of it, of course, is equal to twice. Um the, uh, twice do it like this sweeps, so it is equal to ah twice the length or twice the R squared, minus X square to the 1/2. So so it&#x27;s equals twice the length are placed the height of the curve at this point. So, um, if we it&#x27;s so so it&#x27;s twice dis length here, which, of course, is just, uh, just, uh, the square root here, um, and ah, um, and then pee of acts, of course, being the oops. Maybe do it over here. P of acts. Um, please of X. Well, that&#x27;s the length of P of X, Of course. Uh, it&#x27;s to, um P of X. Here is the length from here to here. Uh, so this is my p of acts, and by inspection, I guess you could see that, Of course, that&#x27;s equal to just being are minus. Well, that&#x27;s my pee vacs and my h of X. Now, the next step is, once I have these guys, I put it in the in a row and evaluate really, really the, um, the big step is right here. Once you have this than the I guess the next step is to set up of the inner grope. Um, okay. And since I&#x27;m using shell message, then automatically, I have the two pi. This is from the, uh, shell message. So this is really the shell nested anyone? No. Hey, that&#x27;s because I&#x27;m using a shell method now. The R minus sex, of course, is the P. That&#x27;s my That&#x27;s my p of x o. Um Oops. Maybe I do this. Okay, so, um, this is my pee here, folks. Come on, little baby. Okay, so it&#x27;s not behaving like it should be, Um, So this, of course, is P. And, um Then, of course, the rest of this is just the h. So this is actually a so that&#x27;s h. Um, so this is a TSH. I&#x27;m now on. We will meet when we of course, the the evaluation. Um, let&#x27;s go back to, uh now the A should be the point of the domain. I guess they had the first plant, Which years are and, uh, furthest point. It&#x27;s negative. Are so my inner girl should go from our two negative are, um Okay, so this is being my okay. And once I have that now, really, at this point, I&#x27;m actually done pretty much because what&#x27;s left is just the evaluation of the integral, um, and of course. And now this. I don&#x27;t know. This is kind of just write it out here. So, um, this here, uh, just separating the inner Groll, um, into this. And it turns out that this this integral here is actually zero. And this one were given in the problem as equaling two. This is actually equal to I just, um, factoring if I factor out the two times big are, Then I get that this is equal to, um for pie. Uh, or pie Big are times. Um, now the given from the problem. Ah, that the the inside inner Groll is sorry. Move out the Patronus into four, huh? They are the integral. Hear this? I mean, you can really you can really look at it. This is actually the area of the half circle, so it&#x27;s actually equal thio of radius. Little are. So, um so that&#x27;s actually and, uh, pie times Little are to the second divided by two. And of course, this is just equal to two pi squared. Little are squared. OK, so it&#x27;s sorry I forgot the big R. Um yeah, times big are times they are. Okay, so that&#x27;s really that&#x27;s really all there is to it. The trick is in the problem again. Is getting the diagram right once. Once you get these diagrams right, then you&#x27;re then you&#x27;re really you&#x27;re You&#x27;re really pretty much there. So if you can get the diagram right and everything, then you know, um, evaluate what? The proper Ah, the proper method to uses and, well, a good method. And then, based on the method, you evaluate what your various functions are Are um Okay, so, uh, that&#x27;s it for this problem. Thank you very much.

Sahil Patel · Answer

So here I&#x27;ve gone ahead and taking one cross section and found the quaking for it. Why? Square plus minus big or squared equals R squared and sold it for X. Now we have two equations here. X equals R plus Reminds the spirit of our square minus y scored. So the red one is gonna be a soc with us and the blue of the minus and I&#x27;m gonna we take the integral I&#x27;m gonna call this one x plus. It&#x27;s when x meyers What&#x27;s he taking? Explosive squared minus X minus squared. Rotate around the y axis. Yeah, you Why? And it will be multiplied by pi. And I think we can make it will be easier for ourselves and notice that symmetrical and just take it from zero to our most wide by two. So then I want to do a square These What&#x27;s the thing? That I want us to notice that a minus b Wait a plus B where my a minus b squared. It&#x27;s just for a B because the a score of the B squared will cancel out. So thank you. Just aargh to pie zero are or are r r squared minus y squared you. Why now here for our is a constant so we can take it out and put in the other side of integral Excuse those a pie four times in a row from zero to our of spirit of r squared minus y squared d y. Now I would have backtrack, and it&#x27;s sort of solving this integral to get our answer. I want to do it another geometrically. So the area of this circle is high. Little r squared. And when we rotate around, um, the Y axis, let&#x27;s just look at its center. Its center will create a circle with radius are So the circle has this your conference of two pi r They&#x27;re really rich just taking this one circle on dragging around two pi r times. So I say that our value it&#x27;s gonna be equal to that area times is your conference. And if we look at a sort of like with the polar argument, we can see that each angle from the center the area is pi r squared. That&#x27;s just gonna be two pi r as this, your conference dragged around. So I would say that our answer is two pi squared little r squared cigar, and that is what this interval would give us if we read about

Problem

Volume of a bowl A bowl has a shape that can be g…

Problem 53 Hard Difficulty

The volume of a torus The disk $x^{2}+y^{2} \leq a^{2}$ is revolved about the line $x=b(b>a)$ to generate a solid shaped like a doughnut and called a torus. Find its volume. (Hint: $\int_{-a}^{a} \sqrt{a^{2}-y^{2}} d y=$ $\pi a^{2} / 2,$ since it is the area of a semicircle of radius a.)

Answer

$2 \pi a^{2} b$

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$2 \pi a^{2} b$

Thomas Calculus

Heather Z.

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Integration Review - Intro

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George B. Thomas Jr.Thomas Calculus

Heather Z.

Kayleah T.

Caleb E.

Joseph L.

George B. Thomas Jr.

Thomas Calculus