a-torus-doughnut-find-the-volume-of-the-torus-formed-when-a-circle-of-radius-2-centered-at-30-is-rev

Question

A torus (doughnut) Find the volume of the torus formed when a circle of radius 2 centered at (3,0) is revolved about the $y$ -axis. Use the shell method. You may need a computer algebra system or table of integrals to evaluate the integral.
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Dorcas Attuabea Addo · Accepted Answer

yes, so in this example would want to find the volume of the exerts when giving a sickle with a radios and a ST er. So giving a cycle off center. That&#x27;s it. This it&#x27;s I saved. Uh, so we have a saint. The sea. Let&#x27;s say this is the center of the same size three. So and I read, use is soon and we are working about the why adds is about the way exist. We know the equation off its cycle, so we think the equation of its second by consider they creation off a sickle. The question off a cycle we have seen. It&#x27;s his words Exc my let&#x27;s see square legs. Why squared equal two outs? Great Are is a real sea is the same thing. So you realize that we have X minus three squared flats y squared equals Who full So this is the equation. Our F off eggs. We are looking at the late see. So from here we can make why the subjects lets you want to find ffx. So if I make why so I have wife squared two vehicles for my NIDs. It&#x27;s when it&#x27;s three straight. If you find why you realize that why is going to be Bloods? Oh my neck&#x27;s And this is the hides the show hides we are looking at, so have full minus X minus three. Swigged hockey. So what does it mean? Say&#x27;s It&#x27;s a Turow&#x27;s. You realize that with its roads, they say, If I have fades, you have the upper and the Luna. So you have your inner psycho and you&#x27;re out, sir, cycle it just like you have to semi circles and each off that is words is this that we are looking at. So since I have to up, that means some grace it gets so far they show high you and I&#x27;m going to have to off for my needs. Eggs minus three straight. So he gets to off debts close. Yeah, Quincy Woods A. To its. So with two of that, I can then rice my find my full. You see, if I was writing for the volume, you realize that&#x27;s out for you is going to be quote to be is going to be. They see graph ordains ever AIDS to be You have soo pie eggs, signs you show height and now they show how it&#x27;s what we have. We have to squares of words X minus to B squared. So this is my next class waas plus our full that is from here the words be eggs So this is the volume we are trying to find. How day do we find the in seva A and B So you realize that if I want to find the interval so for the things of ah, for the inside of ah ads why he called to so immediacy a having words x minus three squared equal to full If it&#x27;s so basically I&#x27;m getting a quadratic equation and you find their roots off eggs We should give you X is going to vehicles one and fight. So then we know what the inter va looks leg. So we have a need to develop one in five so we can then writes the volume the so they we have be so V is going to be a kilometer right? It&#x27;s here. So V is going to be day in siga from 1 to 5 I have four pi eggs full high eggs. Then you&#x27;re scares off words full minus x minus would be we would squid the eggs because four pi is a Constance Constance. Here. You can&#x27;t bring your constant out so that I can have four by you see once. Five. I have It&#x27;s have eggs. Screws off for my knees. X minus was three squared the x Not This is the insane Crowley wants so so less to Alexa Substitution. So let&#x27;s make legs. I want to do substation here, so that&#x27;s I kind right. That&#x27;s so from here. So now let&#x27;s consider this in seagrass. So let me write this in. See gravity. Let&#x27;s consider that one alone in Seoul. See if I can see that. That&#x27;s unseeing eye. Let&#x27;s see one. Sue five in. Siegel wants to buy eggs full, minus eggs minus three. Squared the woods X Very good. Let&#x27;s let&#x27;s say we want to legs. You vehicles Woods X minus woods. Three. If you tickle two X minus three days implies that our eggs is going to be Paul&#x27;s words. You blast was three, so they I can&#x27;t do this substitution here, so that&#x27;s from here. You realize that from this in cigarettes and I&#x27;m going to have any see graph, they see one soon five I have was you last three. You blast three name I have You&#x27;re scared is four minus was used. Trade was the would be you Ok, do you? Because we changed for the X. If you find you the x year, you the eggs here it&#x27;s people to one. So the X year it&#x27;s equals words. Do you Okay? So that is the substitution we have the with a substitution. We can then so full. So let&#x27;s expired the integral We help if I expand this in Seaver If I explained this and see grand you it&#x27;s going. So give me by expanding. See gra then I&#x27;m going to get this is going to be this Instagram is going to give us Zika from minus 2 to 2. Don&#x27;t forget it&#x27;s when you make the substitution your boundaries we change us will for eggs to be equal to one You realize I see I get seeing you two B minus two then for X to be quoted five guessing you to be what&#x27;s equal towards soup. Immediately do substitution. Your intervals change as well. So this because they get sick too. And sue they So from here you realize that&#x27;s how I always have the cigarette expanding. This would give me. You have full minus use Squid last three. Scary. So four minus you. You sprayed the words the U. So if I have to lie that some roots Alexey Applying the some applying the some You realize that&#x27;s what we have. We have in C gra minus two exits to have you and I have for my needs. You suede Do you? Class three the recent Constance I can bring three outs in C gra minus 2 to 2. I have woods four minus was you, skway be you useful to you? So you would want to solve. Take your time a So the insignia you want to. So for this and so far, that&#x27;s us will and you solving for this one you will need integration by parts to solve. And this with a serious work. He realized that this kind off integration needs and computer algebra Two So for that so you could use a mathematica are really soft way to make it easier. So applying integration by HUD&#x27;s to So this will go towards zero. So then we&#x27;ll be left with this parts off the integration. So So we are now looking at three in siga from minus two to sue squares off folk minus you square the was you So let&#x27;s so we want to apply trick substitution So let&#x27;s say we have play Trigg sub sushi. If you have a place trick substation wants to make you to be equal to words to sighing V So that&#x27;s view. Do you Devi here he&#x27;s going to be would sue us v the u nuvi Okay, so that I have my new vehicle to to cause V the words Devi. So if I know this, I can&#x27;t do the substation but don&#x27;t forget any time reduced, we make a substitution Then we have to work on our boundaries as well so that if I work on my boundaries, you realize that So if I adjust the boundary if you adjust, they&#x27;re bungee, they&#x27;re bungee. You realize that for you vehicle so minus two we are getting was vey difficult minus pi on Sue. If you do that substation here, their vehicles to this implies that v, it&#x27;s equals were high onwards soon. So we have a new woods, a new a boundaries that is pi minus by own sue and woods Hyo Suk in pilots. So with this, we can do do the substitution. And so full the in Seaga. So now we have. So you let me write this. So just for the sea grey Loon So many rights for this insignia minus to sue. Don&#x27;t forget I still have my full use Quigg the u with a substitution. I end up with the zebra minus by own sue I own sue from so I have woods. Mine is full minus. So sign V, which is my use squared here then you do you? The U. S was sued because v the Woods Devi If you simplify, then we are gets Indian cigar for minus my own soup I won&#x27;t sue you Simplify the whole of these and you get say, four calls Quaid v the v Huh? Full codes we really be so they can look eyes. So from here No, Let&#x27;s simplify the integration sido forgets from. So this is what we initially had initially had three minus by on sue uh to for cause squared v the b Okay, which is equally equal. Two words. So this it&#x27;s equally because your four times three wishes so because it&#x27;s a constant. So they have Swope from call squared. We we would be This is spy on Sue minus high or soon. But don&#x27;t forget, I can express call screen so I can express cause Great us So let&#x27;s see. I can next race course grade course Quaid v c vehicles The one Billa&#x27;s cause Su v uber soon. So that&#x27;s from here. I can therefore right? It&#x27;s us. I have the eggs igra so off, Watsu, If you do this substitution if you do substitution day, you realize as you have. If I do myself solution, then I have Okay, so we&#x27;ve myself sushi. This is what I have. I&#x27;m going to gigs so off Salami writes from here. I have to pull over at Sue my Ziggo from minus by Own sue Hi own too. This is new More. I have one glass class cause spray caused to be the woods TV. Now this is what I have. Okay, So you you realize a game dies if I find the NC gra and if I find a cigarettes so this is giving me a six, then I have I play somewhere here. This is going to be words the that is minus by on sued then high on to Why? Because if you find integration for this pots and you put in the boundaries this buds who go to serve so we will be left with that pack. The one pass which is this? So this is going to give me six science words high, which is called six high. But from where we started for me say that volume what is equal to words for Hi The constants we had for by signs that NC gra we saw which is with six high. So we end up with words sweating See for I scraped. So finally this becomes words our volume for the zeros. We, uh so being full. So we use the shell method Sue, consider please. Which needs a riesgos computer algebra to do.

Tp Sarathy · Answer

problem. 71 were given a circle in the explore plane, but center tree zero radius too. And this is located about the Y axis and a full revolution and were asked to find. And what it produces is a is a donut shaped ah structure a second. And you can refer to the book on that. And we need to find the volume off the donor or on a technical term for this for such a Donald Cheapest bullet for volume of the tourists. What we need to find. Okay, this is a very good problem. Very good exercise in integration and visualization, conceptually conceptualization of the problem. And we are familiar with one invitation off a solid object. Ah, it owned the access. Oh, every except that we have ghost figure in this case. And so let&#x27;s see what that&#x27;s about. I would, and there&#x27;s public different ways of doing this part and the thing that immediately strikes to me since we&#x27;re looking for the usual radius and the access of a patient which is defined. So the first thing is to point of radius, and it&#x27;s really so if you look at it, let&#x27;s I would start with the equation. The creation of a circle. I mean, that&#x27;s a good starting point. West of a circle we know is those as X minus h the whole squared plus why minus k. The whole square equals R squared, where each case, the center An artist, of course, radius it when this case would be given this three zeros descent. So which means it&#x27;s equals three and K equals zero, so we can immediately write down some. An expression for not a question of the circle is X minus three the whole square. That&#x27;s why minute zero, which is the stamp, equals R squares for four, you can see X minus three. The whole square equals four minus y squid and so X minus three. It comes, uh, plus minus credit off minus y squared. We automatically right, because we can take the yes, and we&#x27;re square this same of explain it. You get the same result. So we have two values for expense able to this. In other words, X equals three bless minus squared off, four minus y squared and excess. Really, that what we&#x27;re looking for this excess that distance from the x axis and it makes sense. There&#x27;s not for this figure that we have actually two values of X indeed forgiven. Why we have that those thanks and then ordered this. That&#x27;s your second eggs. Okay, So and and you can see we can identify that this distance the distance with white with the second X value is the outer radius and that the 1st 1 x one value would be the inner radius. And so everything is looking, uh, it&#x27;s taking making some sense in right now because we can no map ourselves to the to the integral equation. B equals integral city because it&#x27;s no why High times let out be off. Why the whole square minus you? Why the whole square? The way This is the out of a tedious and your voice the inner radius. And we know from what we put the figure that ah, we have to London&#x27;s. Here we have. We have x equal honey, less credit of four minutes y squared and that&#x27;s X one and X two is three minus squared off four miners. Why square, Really? You can see that this is a larger quantity, so that will be out of videos in other words. This is the old boy, and this would be the cute boy. So that&#x27;s good. And see the D the limits in the Y direction. So the radius is still so the limits in the Y direction are This distance here, we know, is because really assist to is this quantity is minus two or two in magnitude. And this money here, this plus two. But this is three plus two. In other words, that&#x27;s three come on to and this is three comma minus two. All right, so that&#x27;s great big art. We got everything, but that&#x27;s right. The volume is now becoming a spy. Integral minus 2 to 2. Be of why the whole square. But that&#x27;s this quick. They plus this sold with okay, hope with me. It&#x27;s three plus sweated off, four minus y squared. The whole square, minus the minus four minus was quit. The whole square you want because that&#x27;s that&#x27;s you know why. That&#x27;s be of what again. And so let&#x27;s see we have It&#x27;s a little messy, but if you if you can recognize that distance of the bomb a plus B, the whole squared minus a minus bi, the whole square it might be might make things easier. So we have a plus B, the whole squared minus a minus bi the whole square. And we know that the A squared B squared is quite discoverable. Cancel out and we&#x27;ll be left it to a B minus. Minus two. A. B. That&#x27;s for a B. Where is three and B is this Well, let&#x27;s let&#x27;s by process of Williams, then I Times Reminder started to or times is three now the baby is later Bill. All of minus y squared the one so which is equals. Drill by minus two before things 12 out squared off. Four minus y squared. Be what? Okay, I want to see this when it&#x27;s to see off one minus boys. Quit I wanted to do. Is this to square it off? One minus y squid. But they have a four year. The best thing is to make the white also a put off four by squatting between believed to somebody in the white. And then so you should think of show You probably total when your people go, uh, sort of this using use substitution that the this is slightly different because you substitution matter and make a right. You know why we need a 400 square it. So we&#x27;re gonna have to and all this legal. I&#x27;m sure your daughter scientists are close until you can use either of those. So this is a perfect solution that big we have four times one minus sine square. Teacher becomes cost Quetta and squared. A discredited becomes school science data and they think simplified. Look and get this first things is Ah, let&#x27;s see. So that doesn&#x27;t important. Did you? Today that&#x27;s so important. Substitution. We&#x27;re gonna effort of this often. Well, let&#x27;s see. Take a d y de y becomes so no sign data dated one and then another looseness that, like was Thank you by swims France quit data and so squared off four minus west Quest become better four times one minus sine squared Makes it too. We&#x27;ll sign. See why? And this one the main eternal tum. So this is now we&#x27;ll make squared off four minutes y squared. Rewrite this as as data and the wives right has data defeat. But I&#x27;m used these to lead with William Now becomes high times didn&#x27;t approach to except well, we had a 12 point? No, by I&#x27;m Brooke of these, but two times cause antique active times goes on better for, uh, actually relate that dilemma simply at times to call nine. Okay? Or take out one of the tooth and use one of the truth by the outside. This Yeah, gosquared, you know, 80. Okay. And all this, actually, we&#x27;re gonna just carried limits Limits should actually change because we changed over to Dana, but for timing and will keep it. They have to In terms of why? Because I want to come back to white then. But 24 by times. This and, uh, let&#x27;s see, this is one. So they&#x27;re continuing along carefully. Do it. This becomes integral. They wanted us. You go. And integral. Of course, I&#x27;m you guys sign, but to Okay, so I&#x27;m gonna come back. And I said, oh, distant. And really, I think maybe Ted quantity. Okay, I&#x27;m gonna date and signed to them, but we need in terms of white, but that&#x27;s the limits. Are native to let&#x27;s find out of work for this means how do we come back to white? No, that&#x27;s if data is this, Um, a couple one the rumor substitution waas a substitution. Why equals sign? This means sign Tito is why, over two or nine hours, Sign in words. Why? Over a bust in here I would sign to theater y equals scientist. You know that Find Dubica sequence Doom&#x27;s equals sign. Okay, now to scientific elderly Vinos Why had a member of your work? I&#x27;m trying to come back to White completely, but to scented awas. Why, that&#x27;s order no substitute and course I&#x27;m Tater because because poor sign Nice. I waited on one minus and square leader This sequel squared off on minus sign Taito squared So scientist is way over to from girl The science squared will become That&#x27;s quite a while but we just quick Okay, well, now, which means Sant&#x27;Egidio. Let&#x27;s go back with the same period. Which is why times course 70 Devic is equated off one minus. I spent over four. So let&#x27;s see. Basically this this ex Christian here What? We are trying to simplify and evaluate all that incomes of theater. There&#x27;s no back in terms. So what? Because they didn&#x27;t know this was where would and signed to data. Let&#x27;s not forget the two. There is is this guy, which means we can write. We can like the volume is the setting 24 by Write that down by Okay, And now let&#x27;s strike this ass for negative 22. Bigger was signed in was why over two then plus beheld that until teacher divided by two I won&#x27;t write it by trio signed today Doha&#x27;s This one. Why times laid off my squared, work emanating looking thing on the part of what we&#x27;ve been saying all along. But but then you can see this. So now we happens. Plug in very to plug in the values for too negative too. So let&#x27;s see an important too. This will be before by I&#x27;m find inwards going to less times two times one minus or we&#x27;ll therefore Wow, that drops are, doesn&#x27;t it 11 minutes 10 so of ghastly. I hope so. Okay, that&#x27;s for any of that&#x27;s what plus two, then minus the same people are native to this will be sign in worse negative, too less same thing you&#x27;re half. Why times one minus when you could negative too. It&#x27;s the same thing. This is negative, but it still crops up. It becomes So it&#x27;s hard. Negative two times zero. So that&#x27;s gone. So this this I&#x27;m in worse. 2 to 1, right? So that seeds 24 times by mining was one. Well, let&#x27;s carry that day and then minus, you know, signed was minus one. But we can in your some nice comedian values for ah, well, let&#x27;s say I mean was oneness by, would you? And sending was minus one is one, and by over tourism becomes 24 times by times by over two minus finest by over. It&#x27;s just I So you get a total value off 24 times I squared. Let&#x27;s see. In other words, volume off the owners. Now our problem is 24 i squid. No, this is all using calculus integration rotation. Well, it&#x27;s about access and a whole bunch of things. But we had to read very carefully in do all these things, not mistletoe thing, and come up with this swelling. But I think it makes a lot of sense to make sure that and competitive what the standard formula for, or the world him off the tourists. And we know that I don&#x27;t know how many know this, but the volume is there is a standard formula that he can refer to, um for the volume or for tourists. It goes this So this 10 books will tell you that the one remove returns mint cross section radius one are and And what do we call it? End The year and large radius are rotational videos. Oh, maybe. But clearly my once I vigil favorite figure. In other words, if you have if you don&#x27;t like being a donut like said he had a and then much we have been ultra stream. Okay, so let&#x27;s say that with this radius was are the in the radius So we have ah, our section. So can you make a section? That radius was our and we had a bigger this, uh, the distance from the center of the donor to the center point of the them there and back waas that distance waas upper case are then the wall you miss you owned by one is going by two pi squared r r squared given by two My right are our script that actually our our script in allergic so you you only square the smaller radio but that&#x27;s the standard formula for the volume of for tourists. So we should all right, It&#x27;s a good idea to like in the values for you know, all this. You know, we have this value s radius of our circle was given us s a to And this the distance from the center toe the ah central, um of the circumference was given This s three So capital of the street. But it makes apart according to the our problem This this son is rotated on the back like that That radius is three indifferent of the book for glad if I what I&#x27;m I&#x27;m saying But this value a street of a kinetically plug in the these values and evaluated and find out It&#x27;s one you must two pi squared r squared is four all the street and so that gives you a value of four times. When do four i squid What had become of it? You see your long and long world calculus matters of integration and think is identical. 24 place quit. So it&#x27;s just the perfect. This is a wonderful example. Great learning, uh, exercise kind of long, but a wonderful exercise that tests your knowledge and, uh and ah, accuracy. You&#x27;re integration skills and, among other things,

Clarissa Noh · Answer

Hey, it&#x27;s clear. So when you right here. So we have a circle of a center with three common zero with the radius of two about the light access and were given X minus three square plus y square is four with the given information, so we know why is equal to plus or minus square root of four minus X minus three square from one 25 So when we look at part A, we&#x27;re going to say F N g or continuous functions where F of X is bigger than G of X on an interval from a three B. So let&#x27;s state the region is buying the curves of what f of X and G of X, so it&#x27;s between X is equal to a and X is equal to be. So when we look at the volume, we get the integral from A to B, the shells or conference, which is two pi X multiplied by the height of the shell, which is f of X minus G of X VX and we know co sign. It&#x27;s bigger than zero or equal to when the interval is between zero and square root of pi over two included. Then we know that the volume is equal to the into girl from 1 to 5 for four pi X multiplied by square root before minus X minus three square DX, and this becomes equal to 24 pi square. And for part B, we&#x27;re looking at the washer method, so you look get the volume. This is equal to the integral from a to B pie of F square of why minus Jesup y square d y. This becomes equal to the integral from negative to to to for two pi three plus square root of four minus Why square square plus three minus square root of four minus. Why square square d y. So when we look at part A, we know that the volume is equal to 24 pi square for part C.

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.

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.

Carson Merrill · Answer

So for this problem, Uh, we&#x27;re looking for volume, and it&#x27;s the volume found by rotating a circle around the y axis. Um, so this is going to give us what is known as a Taurus? Um, and we&#x27;ll have X minus the radius squared plus y squared equals r squared. And what will end up using is the washer method to solve for X. So when we saw for X, we get that X is equal to R plus or minus two square root of R squared minus y squared. We&#x27;ll have the positive being the right half and the negative being the left half so we can use symmetry. We know at least symmetrical. So we will integrate from zero to our and just double it. We know that our radius one is going to be the positive, and our radius two is going to be the negative. So we&#x27;ll subtract inner from outer, and we get that the volume is equal to two times the integral from zero to our because we doubled it, remember? And that can be times pi or of pi times. The first radius, which we know to be R plus the squared of R squared minus y squared and that value is going to be squared minus. So the outer ar minus the square root of R squared minus y squared and that value squared. We integrate with respect to why and what we end up getting when we simplify is eight pi r times in a row from zero to our of r squared minus y squared dy then for part B, we&#x27;re going to make use of solving. When we saw for excellent square both sides, we get the equation of a circle. So what we know is that X is representing the right half in quadrants. One in four, um, the integral limits zero to our, uh so that&#x27;s half a circle, which is 1/4 it&#x27;s half of the half circle, so it&#x27;s 1/4 of the full circle. So what we can do is replace the integral with the area of a circle, which we know to be pi r squared and then multiply that by 1/4. So we&#x27;re going to plug that in here, and what we end up getting is that this is going to be eight pi r times 1/4 pi r squared. So that&#x27;ll give us our final answer of two pi squared r squared big are

Problem

Different axes of revolution Suppose $R$ is the r…

Problem 69 Hard Difficulty

A torus (doughnut) Find the volume of the torus formed when a circle of radius 2 centered at (3,0) is revolved about the $y$ -axis. Use the shell method. You may need a computer algebra system or table of integrals to evaluate the integral.
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Answer

$16 \pi^{2}$

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William Briggs, Lyle Cochran, Bernard Gillett

Calculus for Scientists and Engineers: Early Transcendental

Anna Marie V.

Caleb E.

Samuel H.

Michael J.

Integration Review - Intro

Definite vs Indefinite

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$16 \pi^{2}$

Calculus for Scientists and Engineers: Early Transcendental

Anna Marie V.

Caleb E.

Samuel H.

Michael J.

Integration Review - Intro

Definite vs Indefinite

William Briggs, Lyle Cochran, Bernard Gillett Calculus for Scientists and Engineers: Early Transcendental

Anna Marie V.

Caleb E.

Samuel H.

Michael J.

William Briggs, Lyle Cochran, Bernard Gillett

Calculus for Scientists and Engineers: Early Transcendental