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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.(IMAGE CAN'T COPY)

$16 \pi^{2}$

Calculus 2 / BC

Chapter 6

Applications of Integration

Section 4

Volume by Shells

Campbell University

Baylor University

University of Nottingham

Idaho State University

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

24:22

A torus (doughnut) Find th…

04:03

A torus is formed when a c…

11:17

A torus is generated by ro…

07:14

Use cylindrical shells to …

02:52

(a) Set up an integral fo…

The volume of a torus The …

03:07

Use the theorem of Pappus …

08:49

In Exercises $51-54,$ use …

03:47

A torus is formed by revol…

05:21

Volume of a Torus In Exerc…

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's it. This it's I saved. Uh, so we have a saint. The sea. Let'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's his words Exc my let'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's when it's three straight. If you find why you realize that why is going to be Bloods? Oh my neck'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's It's a Turow'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'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'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'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'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's so basically I'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'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'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't bring your constant out so that I can have four by you see once. Five. I have It'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's make legs. I want to do substation here, so that's I kind right. That's so from here. So now let's consider this in seagrass. So let me write this in. See gravity. Let's consider that one alone in Seoul. See if I can see that. That's unseeing eye. Let's see one. Sue five in. Siegel wants to buy eggs full, minus eggs minus three. Squared the woods X Very good. Let's let'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's words. You blast was three, so they I can't do this substitution here, so that's from here. You realize that from this in cigarettes and I'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'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's people to one. So the X year it's equals words. Do you Okay? So that is the substitution we have the with a substitution. We can then so full. So let's expired the integral We help if I expand this in Seaver If I explained this and see grand you it's going. So give me by expanding. See gra then I'm going to get this is going to be this Instagram is going to give us Zika from minus 2 to 2. Don't forget it'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'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'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'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'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's to So this will go towards zero. So then we'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's so we want to apply trick substitution So let'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's view. Do you Devi here he'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't do the substation but don'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're bungee, they'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'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'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'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'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's equally because your four times three wishes so because it'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't forget, I can express call screen so I can express cause Great us So let's see. I can next race course grade course Quaid v c vehicles The one Billa's cause Su v uber soon. So that's from here. I can therefore right? It'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've myself sushi. This is what I have. I'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.

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