cylindrical-shells-in-section-62-we-learned-how-to-find-the-volume-of-a-solid-of-revolution-using--2

Question

Cylindrical shells In Section $6.2,$ we learned how to find the volume of a solid of revolution using the shell method; namely, if the region between the curve $y=f(x)$ and the $x$ -axis from $a$
to $b(0 < a < b)$ is revolved about the $y$ -axis, the volume of the resulting solid is $\int_{a}^{b} 2 \pi x f(x) d x .$ Prove that finding volumes by using triple integrals gives the same result. Use cylindrical
coordinates with the roles of $y$ and $z$ changed.)

Yiming Zhang · Accepted Answer

so the proof. This is the start with a substitution for cylindrical coordinates. So for cylindrical, we have X rays or comes on theater. Why&#x27;s what Selvan Z is for Cynthia? That&#x27;s over. Substitution for cylindrical coordinates. So hence the bull. We&#x27;ll be equals shoe. So we&#x27;re gonna expressed of all, um, in cylindrical hornets. Witches are the hors de theatre B Y. So you know, we&#x27;re setting which just up flogging the values. This is a to B 0 to 2 x and zeros out of our or the white, the state of the or Okay, So the evaluation of this since the girl is finally to buy some say to be or f or the or, huh? So what is the same thing as to part from a to B X of That&#x27;s the ends. Still, that shouldn&#x27;t coordinates. We can prove surgical internal will be transformed into a simple one. Variable. Devon&#x27;s into group. That&#x27;s it.

Ayush Naidu · Answer

tells us that the show method was used to find the volume of a solid generated by region are when revolved about the Y axis. Also given that region are is located in the first Quadrant and is bounded by the graphs of two minus X squared and y equals X in the close general zero comma to With this information of drawn in the little graft buster out this problem hard A assets to identify the radius of a cylindrical shell at a point x in the u nivel. In order to do so, we&#x27;re going to use terms of X throughout this problem because we&#x27;re dealing with the vertical axis of revolution. So the radius, which is R of X, is the distance between the point and the access of revolution, the point being the point where the cylindrical shell occurs so our vex would be equal to a point. X minus 06 Taxes Revolution is X equals zero, which is equal to X. So the radius of this celery Cochell Zac&#x27;s be assets to find the hype of the cylinder shallot a point X so h of X, which is the height with the length of the rectangle is equal to the topic. Question one is the bottom equation in region are because we&#x27;re dealing with terms of X. If who was terms of why we would go right minus left. So since its top minus bottom, it be two minus X squared, which is the top equation in regional minus X because Y equals X is the bottom in question. So there&#x27;s your help. Part C asked us to find the volume expression for this solid when it&#x27;s revolved about the Y axis and it&#x27;s generated by the shark. So we know that the generic formula for value so the generic formula is you go to two pi times the integral from eight a b off the radius times the height times the change in X, which is denoted by D. X. So we know this is the generic formula. If we apply to this context, it be equal to two pi two pi times the integral from zero to to both the radius which in this case is X times tu minus X squared minus x because two minutes exclude minus x was the height that we determine in part b times The change in X, which is DX because we&#x27;re dealing in terms of X. So this is our final answer for part C.

Bob Monday · Answer

were given, Um, problem where we have a cylindrical show. So something that looks if we were to graph it like this with the inside kind of out of it. This is a big old hole in the middle. And we&#x27;re told that for this show, the, um inside radius is given by a The outside radius is given by B and then the high other. The head of the shell is given by Sea Knight C and we&#x27;re told that we can You can write the function f of X y Z in the form lower case f of our plus h of the were here when we&#x27;re talking about our, um and see these air these air cylindrical coordinates. So those were the normal formulas were also given that if we have F Tilda is the anti derivative, um, well, say I&#x27;ve till the end age. Still, though, sorry, F Tilda Age Tilda are anti derivatives of f and H, respectively. And using that week it we have some information here specifically that the integral from A to B the same they MB that we have here for the inner and outer radius of the anti derivative of f with respect to our d. R is zero and then we also have that the ah anti derivative of H zero is equal to zero. So we&#x27;re going to do with this is we&#x27;re going to find a formula for the integral, the triple integral of B. We&#x27;re going be here is this, um, cylindrical shell of f of x y z um Devi and we&#x27;re gonna use the information given to us to find to find a formula for this. Okay, so the first thing we can dio as we can we can change this to cylindrical coordinates. So if we do that, then we can write our triple integral should actually pretty easy. We go from 0 to 5. That would be our data. Our radius it goes, since we&#x27;re looking at the region between and be then goes from A to B and then the value of Z goes from zero. And since the height of see we&#x27;re going to see from here, we know that we can rewrite our function f in terms of our NZ and so we could do that. But when we change to cylindrical coordinates, we also have to multiply by our it&#x27;s actually would look right at this one. Well, multiply each of these individually by our our time f of R plus R Times agency and then we go backwards in the order of the Integral. So the first or the inside Integral is for Z Super easy. And then we put d r and then data. I know we&#x27;re just gonna were gonna calculate this, and we&#x27;re gonna use the given information here about the anti derivatives to work this out. So the first thing we can do is we&#x27;ll do the integral we respect dizzy and that integral. So we have from 0 to 2 pi from A to B. And then so the entire derivative of that is going to be our FFR Z, um, plus R and then the anti derivative of H. Whatever. That is that we don&#x27;t know. They don&#x27;t give us unless that&#x27;s fine. We don&#x27;t need it from zero to see if the Argies data. So by plugging in, we see we get since equal, Let&#x27;s go down here we have the integral from zero to pi in a row from a to B of. So the first thing we have in our half of our time. See, um plus R h prime or age Still TFC. Now we&#x27;re weak. We&#x27;re gonna attract in playing NZ are zero. But if we plug in zero into the first term here than our f of our time, zero gives zero. And then we&#x27;re also told sexually what we can write out our fr zero. But they were also told that we have our h Tilda of zero here h 200 We were given up here. Zero. So, actually, so both of these are going to cancel it DRD thing. Okay, so now now we can do is we can actually simplify this one step further. Um, since none of these are in terms of data, we can actually write this as the interval from 0 to 2 pi times One of data times the interval from A to B, our FFR our time. See, plus our h primacy d r. We can do this and it&#x27;s helpful because now we see that this is just a single variable for our and the integral From 0 to 51 it is just gonna equal to pi. So So now where should you aware about this integral here and then multiply that by 25 and we&#x27;re done. So in this case, um, we could we&#x27;re gonna Yeah, we&#x27;re gonna break this apart. It&#x27;ll be easier to just look at two separate pieces. So the first part would be from a to B of We&#x27;ll take the sea out there are fr d r plus. And then h h told O. C again is is actually a constant So you can take that out age tilt of C in a row from a to B of our d. R. So when we when we work it out this way, Um, well, we can see is that in this first inner girl, it&#x27;s not necessary obvious what the first step is. But we&#x27;re told information about the anti derivative of death, and so that would lead us to think that this might be a good integration by parts problem. Um, because we can take the derivative, are get rid of our and we actually know something about the anti derivative F so that even though we don&#x27;t know what efforts would still use the entire derivative. So we&#x27;re doing a great effort to get you vehicles are Do you, Nicholas D. R. We would say Devi equals f of r d R and therefore V is just the entire derivative of f of our so that when we were at that integral. So if c times, um, we have our times f tilda of our minus the anti derivative of just f told of our but we&#x27;re evaluating these from a to B. So this is for me to be here, and this is for me to be here. So again, this is only talking about this first inner girl here. Now we can see that this is going extremely helpful because again, the given information the anti derivative or the integral the anti derivative from A to B is zero. So you can actually grow in here. We can cancel that out. And so now we&#x27;re left with is just plugging in Vienna into our it&#x27;s affecting them. So here we get C times B. In this case, f Tilda be minus c times a f. Tilda a van. Okay, so that&#x27;s that piece right there. Uh, and then we have over here. If we look at this inner girl, the antidote artists are screwed over too. So this is going to be hte Tilda of C times In this case, B squared over two minus a squared over two. And now we can put all together. So here we were, adding, so we should bring We should be careful here that, um that this is plus this right here. Tilt of C a times B squared over two minus x squared over two on that. We need to remember that in the end, we were multiplying by. This is two pi right there. So then our final answer would be two pi times bcf, Tilda of B minus a C F. Tilda of a plus age Tilda. See, and be screwed over two minus a squared over two. Now this this doesn&#x27;t match exactly what? What&#x27;s the answer given? But we can see that, um, by the shooting in the two pi into so we can consider, like, to this to be the first part on this to be the second part so we can distribute that into both of those parts. But then we can also factor out to sea in this in this 1st 1 and so we can see if we, you know, multiply this all out. We get to pie Bacon Factory to see from both of those terms and we&#x27;re left with B after all, to be minus f till day plus And if we multiply the two pi and then the 2nd 1 since both these air being divided supposed to be squared in a sort of being divided by two multiplied by two will cancel those out. So we&#x27;re just left with pi h told of C and then just b squared minus eight square. And that is the final answer for for part A for part B, we&#x27;re told, or we&#x27;re asked to use this result to show the following. So for part B, we need to show then if we integrate over the ball, be of Z plus sign square root of X squared plus y squared dx dy wind Easy that this actually this equals six Hi square times pi minus two. Now we can see here that, um that the sea here is like the age of the and then given the extra plus y squared, that sign is the f of our okay. It&#x27;s definitely going way changes since the cylindrical coordinates. We would get this as a function of our It always did. He was just check to make sure that, um, everything all the properties matchup. So the first thing we see here we have to check is that the anti derivative of H at zero gets a serum, which is definitely true because the anti derivative of H is just c squared off to plug ins. Eerie gets here. We also need to check that the integral Oh, sorry. We also need to see what B is here. Let&#x27;s do them. B in this case is given to be, um, the cylindrical shell. Well, just draw the picture with inside radius of pie, so it will look like this, or we cut out the middle. So the inside radius is pie. The outside radius is two pi on the height of it is to, so we need to check them. So, um, you need to check that the integral in this case from pie to two pi of the anti derivative of f of our equals zero. Well, let&#x27;s see your the said so f of Oregon is just sign of our the anti derivative of that is negative coastline of our And then when we do the integral from pie too high of negative coastline of our what we&#x27;re gonna get is we&#x27;re gonna get negative Sign of two pi plus sign of pie. And the important part here is that a sign of two pints on a pie are both zero. So this is just here, which is what we want to be. So this problem satisfies all the requirements. So now we just we just plug everything in and we hopefully get a six pi squared times pi minus two. Okay, so let me Let&#x27;s rewrite the formula down here. So the formula we have for this dancer here should be to pie time. See, B f. Tilda of B I miss a f tilda of a plus pi b squared minus a squared times H Tilda of scene. Okay, so So the answer for integral then our triple integral that we have be for Z plus sign of squared of X squared plus y squared dx dy y dizzy will equal to pi. In this case, C was the height which is to see us to hear. So a is pie V is two pi I see it too. So 25 times Two times two pi times f tilda of be So in this case, that is negative. Co sign of two pi minus pi times f tilda of a. So this is negative co sign of pie plus pi times. Let&#x27;s see here. Be squares off. Four pi squared minus pi squared. And then h prime of C h h Matilda is ah playing to We get four squared over two, such as to. And this should this should be our our ah, answer that we got up here, So let&#x27;s simplify some things here. So we have two pi. Times two is four time. Um, Ms Missing a pregnancy there. Okay, um and then we have in the inside here. So co signer, Two pies. One. So this is negative One. So it&#x27;s negative to buy there. Uh, coastline of pie is negative one. And so then that&#x27;s become positive. One times pot are positive on times. Negative pie. Is that negative pie? Plus, we have pie times. Three pi squared. We&#x27;ll put this the two there. So we have 25 there. So there we have four pi times negative. 35 plus six pie to the third. If we keep on going, we get negative. 12 pi squared plus six pie to the third. And now, Now we can see that we&#x27;ll get the answer they want. Um, if you look at the given answer here, six pi squared up. I minus two. If we factor out six pie from this right here, we get exactly that. We have 65 um, squared and then we&#x27;re left in the inside. Pie minus two. That&#x27;s answer.

Patrick Vaughn · Answer

mhm in this problem. For part A. We want to use cylindrical shells to find the area generated under the curve. Why equals X cubed minus three X squared plus two x Sorry. So this curve is rotated about the Y access and it&#x27;s on the domain. X is in zero toe one. So if we go ahead and plot this graph, it&#x27;s gonna look something like this. So if you find the zeros of this, you&#x27;re going to get so if we factor, it will have X times. So that&#x27;s X squared minus three X plus two. So that&#x27;s X minus two X minus one. So the zeros are X equals 01 and two. So I have zero Here is zero here and zero here. We know the tail features of the cubic the left tail is gonna go down. The right tail is gonna go up. So we know we&#x27;re gonna have some behavior like this. And we&#x27;re looking in this very small 0 to 1 region. So this is being rotated about. And so we have a little that comes back here, and it&#x27;s rotated about it. The y axis. So there&#x27;s kind of our graph that&#x27;s not very good. But so yeah. Okay, so we can set up our cylindrical chels V is gonna be equal to the integral. So the domain of X was 01 And so we have two pi r is gonna be X. And what&#x27;s the height of the shell of going to be? It&#x27;s gonna be Why? So that&#x27;s X cubed minus three X squared plus two x dx. Yeah, right. So we want to go ahead and solve this three have b is equal to so we can bring the two pi out front and distribute that ex love to pie Inter Girlfriend 01 of X to the fourth minus three x cubed plus two x squared and times DX. And now we can use the power role to find that so up. Two pi times 1/5 x to the fifth minus 3/4 x to the fourth plus two thirds x to the third power and we&#x27;re evaluating this from +01 So that&#x27;s gonna be two pi times 1/5 minus 3/4 plus two thirds. Um, And then for the lower limit, we plug in zero. We get zero for all those terms right. So we have a common denominator of 60 or less because we could distribute this tube, so that&#x27;s gonna be too 26 and force this becomes three halfs. Okay, so then we have a common and I&#x27;m in your thirties. That&#x27;s not as bad. So pi times. So we got two times six. That&#x27;s 12. Come on. Uh, minus. So three times 15. So that&#x27;s 45 plus four times 10. So that&#x27;s plus 40/30. Okay, so thats one plus, that one gets this negative five. Then we subtract five from 12. We get seven, so we get seven pi over three. All right? So in part B, we wanna know. Was this better than the disk method? So recall the graph. So in our domain, here&#x27;s one. And here zero. So our curve looks like this. There&#x27;s a little hump. So if we wanted to use the disk method, we would have to find the point where since way would want to convert. So we have wise equal thio X cubed minus three X squared plus two x. So we want to convert this to a function of X. But if you notice on the graph. We&#x27;re gonna need to inverse functions one on the left side of this point and one on the right side. Because it&#x27;s not gonna pass the vertical, the horizontal line test for an adverse. Because if we take ah, horizontal line below this point, we&#x27;re going to see it intersects the graph of two points. So anyone for them? One for there. So it the shell method. We didn&#x27;t have to do any of that. Well, it was almost already set up for us. We just had toe, like, kind of do a little picture and figure out what the height with radius work. Um, so I would say that the show method with a lot easier than the disk method and I completely problem

Bob Monday · Answer

So we are given a cylindrical sell shell Be so slick, something like this and we are asked to integrate over the shell where we&#x27;re told the inner radius of the shell is given by a The outer radius of the show is given by B on the high of the shell is given by scene. We&#x27;re told that the function f of X y Z can actually be rewritten in cylindrical coordinates, Um, as f ar as a multiplication of functions f of our GF Data and H of Z were also given for the information that the integral from A to B of the anti derivative. So again that f till there is the anti derivative of F um, that that anti derivative is zero or that the integral a zero. And so we&#x27;re going to use this information to find the formula for the integral of for B of the function f of X y z. So the first thing we&#x27;re gonna do here as we&#x27;re going Teoh you know, change change this into, um, cylindrical coordinates. Since we know we can write it in this this nice fashion. So if you look at our cylindrical shell we have, um dangle goes from 0 to 5. Data goes from 0 to 5. The radius. Um, since we&#x27;re looking at the inside of the shell is kind of the region we&#x27;re looking at here. The radius goes from A to B where a was the inside to the outside shell and then the height of it s C, which means we&#x27;re going rz value goes from zero to see. And now we can rewrite our function f in the cylindrical coordinates. You know, f of our GF ada h a z de. In this case, we go opposite of the intervals. So we the inside integral is Enciso do DZ every two d r then we to death data. But also remember to multiply when we&#x27;re changing to cylindrical coordinates most five yr. Well, now what we can do is we can actually break this all apart so we can rewrite this as, um the integral from 0 to 2 pi Uh, in this case, geese Ada di fada, times the integral from a to B of our FFR d r and then the in a row from zero to see of HFC do you see? And we can now work this out pretty nicely. Um, the first in the last one. The first interim last integral will be easy Teoh to find in terms of the anti derivative. So we don&#x27;t actually know what the formulas for the specific anti derivative. But we&#x27;re going. We&#x27;re going right, um G Tilden, aged Children to represent those. So let&#x27;s focus on the middle integral here, since this one&#x27;s going to cost us the most grief. Now what we know is that we know that the integral of just the anti derivative by itself zero and so, looking at what we&#x27;re given here, it would make sense to you something like integration by parts. Because here we are. We&#x27;re multiplying by our in the driver of that will go away, becomes one And then we know stuff about the anti derivative f. So let&#x27;s try that we have u equals r Do you hold on equal d are just what we wanted and then we have DV is the f of our d. R and V is now just the DEA anti derivative of us. Whatever that maybe we don&#x27;t know doesn&#x27;t matter. So in this case, what we can do is wait. Gonna rewrite this integral as our Tilda of our from A to B minus the integral from A to b of just f. Tilda of our But we&#x27;re giving up here that this integral equals zero. So that just canceled out. So now we can actually do is you can actually weaken right? Our final And here because the 1st 1 So this interval here is just going to equal G. Tilda of two pi minus g. Tilda of zero, where we don&#x27;t know anything else. We don&#x27;t have to. This is just the fundamental theme. A calculus. We know. Now this middle one is going to be B f. Tilda. Be minus a. I&#x27;ve told of a And then we know this last one here, just like the 1st 1 will be H. Tilda of C minus H. Tilda of zero. Where again H Tilda is the anti derivative of H so and so the answer is multiplying. All three This together our final antihero, v g Tilda two pi minus gee Tilda zero times b f tilde key minus a. After that, a, uh, times h told o c minus age, Tilda zero. And that&#x27;s exactly what we want. Now we have part a for part B. We&#x27;re going to now use this results to actually to find a mineral. So let&#x27;s say we have the triple integral over the domain be of Z sign of square root off X squared plus y squared dx dy y DZ. We&#x27;re going to show that this is going to equal negative 12 pi square. Now, what we&#x27;re told about B is that it&#x27;s a cylindrical shell, just like we had, um, previous previously, um, and were given the inner shell in the outer shell. In this case, the inner shell is just pie. The outer shell is to pine on the height of it is to. So for this problem, a equals pi vehicles to pie and C equals two. So we have to confirm a few things here. The first thing we need to confirm, um, is that this could be ruin in India in the format where I&#x27;m multiplying, you know, the function of the function of our and function if they did together. But we can see again that that&#x27;s true, because here the Z is the HFC the sign, even though it doesn&#x27;t look like it square next group was twice. Court is Are So this is like our f of our, um where you could just repeat this a sign of our and there is no theater. But if you don&#x27;t want one, imagine that there&#x27;s were most find by one there, then the one is our GF data. So this could be written in that form. But the second thing we knew the check. So if we have f of our equal sign of are we all seem to check that the integral of the anti derivative from A to B zero. So the anti derivative here is negative coast out of our and so we&#x27;re doing the in a role in this case from from A to B is pi to pie. Since this is the specific values have here of negative co sign our d r. But when we worked that out, we know the anti derivative of coastline is just just signed. Um, since I took the negative outside the integral, So it&#x27;s negative. Sign of two pi minus sign of pie. But signer 25 cent pie are both zero. So that is your just like it&#x27;s supposed to be. So this fits all the requirements that we need, um, to show this. And so now, now we can just use the formula we have up here. So let&#x27;s let&#x27;s write these out. So we have h of C is C, which means H tilt of Z is C squared over two. So there&#x27;s our H and H still down. Here&#x27;s our F and F dildo. And so then g Zeta is equal to one set me into Ji Tilda of Fada is equal to theta and now we&#x27;re gonna plug those in with our values of A B and C. So then the answer for the integral is going to be So we have g told of two pilot, which is just two pi minus G Tilda zero, which is just zero. So that times now we have B, which is two pi times f tilda of B. So I&#x27;ve told as negative co science negative co sign of two pi minus a which is pi times negative co sign of a, which in this case is still pie. And then that times we have two squared over two. So that&#x27;s H h tilda of to minus zero squared over two. So simplifying this, we get to pie. Negative or co signer two fires one. So it&#x27;s just negative. One negative to pie. There co sign of pious negative one times a negative, is positive, has a negative pie. It&#x27;s just negative pie. And then, um, off See zero script over 202 Squared over two issues, too. And so now he gets two times. Um, well, let&#x27;s let&#x27;s simplify this year. So this is really for pie on the front. So just to five times to and the name of 25 minus pi is negative three pie and now we get the answer of negative 12 pi squared, which is exactly what we were looking for up here. They told vice squad, and so we&#x27;re done.

Sajin Shajee · Answer

Okay. First her party for this question we&#x27;re trying to use Let r be the region battered by the curves. Why? Why cooling? Uh, que the square root of X comma y clung to and exit calling for and which is all continuing the first quadrant of the function. So it should look something like so So that should be contained is starts. They&#x27;re disappointed too. He crashed, going back this point here and then closed Also by X equal for includes the X Quinn war. And why Quinn tears. This is the region where it&#x27;s going So part a a they&#x27;re trying to do What you&#x27;re trying to figure out is the radius of a cylindrical shell. For this is in terms of wife for the region as it goes from as it goes from zero to as it goes from zero all Luigi tubes of for part a frieze is supposed to be all right equals why their part b i supposed to be? Hey is equal to four minus why que two miles wives square? So for part two, see you trying to do the shell method of this one in which Bali missiles equal Teoh integration as you go from zero all the to of two times high times. Why multiplied by four minus two miners. Why, uh, weird there, de why were 1,000,000 also wise according to function wise, equal to to give him a function will be to Main Square O X Thanks for the given language where X is equal for and so far.

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