 Welcome back lecture 10 math 241. We are in the middle of section chapter 6 6.2 on volumes Probably about two-thirds of the way through we have thus far Attacked problems that were volumes of solids a revolution that when we sliced them up They were solid when we had solid disks. We had no volume missing out of the inside Depending on the axis of revolution, but We start with in both cases Pi r squared h for the volume of one of the slices solid disk slices The radius could be either the x value or the y value So if it's the x value Then it's the x value Squared for the radius and if you think about how that would happen Try to get these reviewed real quickly and we'll go on to the third type with which is That solid disk method obviously doesn't work We try to slice it up and look at these slices which are washers But we're having a hard time describing the inner radius and outer radius so we move to cylindrical shells Which would be the third? Kind of attempt to get volume of that region So if you think about the radius of this solid disk You think about that radius actually being x Then what would the thickness or height for that particular solid disk be? That's some increment of y is that correct So I know it looks awkward that we've got x squared dy But we'd have to say what x is in terms of y right make that substitution And if that is the way the disk is shaped or formed Then we would go from some y value to some other y value So that's if the disks are formed in this manner if the solid disks are positioned this way then this radius From the x-axis up to a point on the curve is a y value So the radius is the y value The thickness would be some increment of x some delta x So we would go from some x value to some other x value again It doesn't look right, but we would make a substitution for x in terms of y To integrate the problem we'd make a substitution for y in terms of x so that we could be able to integrate this problem Solid disks are usually the the easiest of the three methods. The other method is If we don't have a solid disk because of the nature of an inner radius and an outer radius we've got a washer Where capital R is the outer radius of that washer little r is the inner radius and Guess we can look at it in that fashion Without breaking this down into into pieces just in general if we have Each of the slices being represented by a washer and this distance would be Capital R and then to here would be lowercase r then we want the amount of Volume in that washer so it can be found in this way It's up to us to decide what is how do we describe capital R? What is the outer radius? Is it an x value on a curve? Is it a y value on a curve? Same thing with the lowercase r. What is it representing if it's a distance from the x-axis over to a point excuse me from the y-axis over to a point That's an x value, right? If it's a distance from the x-axis up to a point that's a y value So we have to pay attention to what curve we're on and is it an x value or is it a y value? I don't think we're in the middle of a problem When we ended class, I think we had just finished a problem using The washer method or the washer method now. Let's go to the third one, which is cylindrical shells What might be appropriate is why would we need This third method so here's a problem and we'll actually do this problem It is similar to one that's in the book, but it is a different Graph and the equation is slightly different But if we have a curve that looks like this Guess we could appropriately call this the nose curve kind of looks like a nose And let me tell you I'm an expert because the Griggs has have some noses Let me tell you it's one way we can tell if they're related to the Griggs is from Indiana. We check out the nose So this is our good old friend the nose curve. Here's the equation of it well if we wrap this around The x-axis and we try to slice through it I'm not going to kind of draw this and clutter the diagram up too much but if we try to slice through this perpendicular to the axis of revolution and Look at the inner and the outer radius We've got an issue on this curve Because the inner radius just ignore this little thing here because what that's the cylindrical shells part But if we were slicing through this and trying to look at the inner and outer radius The outer radius would be here. You would say well, that's a y-value on this curve. I'm already not liking that Because what is a y-value on this curve? What's that forcing us to do? solve this equation for y I'm not really Interested in doing that. Okay, if it can even be done and then you look at the inner radius. Well, what's the inner radius? It's also a y-value on the same curve So we're kind of stuck the the inner and outer Radiation are kind of the same thing in terms of describing them So it's going to be difficult to write them in the form of an equation. So we're not going to use Cylindrical shells unless we really have to but we have to on this problem Because even though each slice looks like a washer the washer method is we're not going to be able to do it So what's it look like? What are what are we going to? Kind of do with this thing called cylindrical shells The name is pretty appropriate because once we decide the little Region of area the skinny little rectangle that we're going to send around the axis of revolution Once we send that skinny little rectangle around the axis of revolution We're going to form one of these which is a cylindrical shell a shell of a cylinder Now how do we accumulate these cylindrical shells? So that we're going to fill up all the volume that we have So let's say the region looks something like this and we want to use cylindrical shells To come up with the volume So we take one of these little skinny little rectangles and we send that around The axis of revolution which in this case is the y-axis And as we do that we form this cylindrical shell Then we would go out a little further Send that around the axis of revolution to form another cylindrical shell go out a little further Take that little skinny little rectangle send it around the axis of revolution and we form another Cylindrical shell so when you kind of put these things I don't know if you ever played with a toy like this when you were a child But they kind of fit right inside of the other right you remember those kind of concentric toys You didn't call them that because the little kid doesn't know what that means But you have a toy that one piece fits inside the other outside the other and you end up with something that looks like this So we can get the volume of something like this by using these things called cylindrical shells You just kind of fit one outside the other until you have something that looks like this If we hadn't chosen our rectangles to be quite so fat We would match this curve quite a bit this region quite a bit better than this seems to do And that's the goal is to take the skinny little regions that you're sending around the axis of revolution Make them as skinny as possible So you can envision these being a whole lot thinner than this and matching this Better than this appears to match it So what we need to do is figure out how much volume is in one of these cylindrical shells and Then start with that and then make it appropriate for each problem that we have so let's take one of these Cylindrical shells So they're going to look fairly skinny, but we really want them to be a whole lot skinnier than that so the reason that we shouldn't or We don't want to at least confuse this with the washer method is that we want this to be So thin that there's virtually no difference between the inner and outer radius So it's not to be confused with slicing this thing up and forming a Washer we don't want this to be a washer. We actually want this to be a whole lot thinner than that So let's say that we have this Cylindrical shell and we know by this time that whatever it is we describe in the integrand We can describe one of them and they're all kind of able to be described by that one description We can add together an infinite number of these things With the evaluation of the definite integral. So that's how we're going to go about accumulating volume if I were to slice this thing just kind of take a Exacto knife and cut it right down there and roll it out Wouldn't we have something? That looks like this use your imagination here Okay on my diagrams. So we're going to cut it Roll it out. And this is what we really have. This is the amount of stuff volume in this cylindrical shell So what is this distance keep in mind that it's really a whole lot thinner than what I've drawn So don't worry about an inner and outer radius What is this distance all the way around here that once we cut it and roll it out? What is that distance? Good, that's the circumference of this Circle right so that's going to be 2 pi r This distance, which is the height of the cylindrical shell is This distance So so far of the rectangular solid that we have by cutting it and rolling it out flat It's 2 pi r by h, but it is 3d. So it does have some thickness Which actually I'll just call that th for thickness So what is the volume of? This particular Cylindrical shell that we sliced and rolled out its length times width times height or thickness 2 pi r times h Times the thickness that's how much stuff there is that's the volume of this cylindrical shell so if we can describe each cylindrical shell in such a way that We know what the radius of each one is we know what the height of each one is we know what the thickness of each one is All we have to do is describe one of them and let the integral calculus add them all up all infinite number of these things together so eventually If we can describe them correctly, we'll be able to get this cylindrical shell Which by the way has a smaller radius, but a larger height, right? Then the next one down which has a larger radius But a smaller height The next one down has an even larger radius and an even smaller height But if we can describe them in such a way that the variables are Represented properly from the curve that we're given this curve we can describe the radius and the height And we can describe the thickness then we're in business we can describe one of them We'll just add them all up with the evaluation of the definite integral So when we have a cylindrical shell problem We'll start off with an integrand And again, there are other versions of this that are in your book and you can choose to memorize those I just kind of like to start with the pi r squared h or pi times capital R squared minus little r squared that kind of Gets me going enough where I can evaluate those things we want a 2 pi r times h times the thickness Now the thickness is going to be either delta x or delta y Depending on how we generated the shells and if it's del x then we'll use x values here And if it's delta y dy then we'll use y values here, but that'll get us started That's the volume of one of the cylindrical shells so back to this example question on that before we leave that Now how many of you before this class had done volumes before? Okay, I can actually see it in your faces. I could have told you which ones had done that because I can see it I mean so you're saying oh, yeah Yeah, yeah, I've already done that So you might want to hide your Expressions a little better from me. I've been at this way too long Now how many of you that raised your hands you also did cylindrical shells you've also done. Okay, so I saw a few Less hands in the air because if something is omitted from volume This is usually the piece that's omitted the cylindrical shells piece So you can do a lot of problems with solid desks a lot of other problems with the washer method But sometimes this is not taught in that section, but it's needed some so let's Let's see how it works all right, so we have this curve and And I'm going to clutter this up and I'll redraw this Actually, let me do this Let me clutter my own drawing up So we have this curve That looks something like this and we're going around on the x-axis. So here's our axis of revolution So we wanted symmetric image down here. So it looks something like this. We slice it up Perpendicular to the axis of revolution and we realize that we've got some missing volume out of the middle here So we think okay the washer method is going to work. Here's where it falls apart So we've got this equation and we need to describe This distance from the axis of revolution up to here because that would be the inner radius now since we went from the x-axis up What is that? When you go locate a point and you talk about its distance above the x-axis you're talking about its Y value right so that's the y value of this point the y value on the curve Okay, fine. I don't want to solve this for why but maybe I could solve it for why and it might be doable Then we say I need to find the outer radius. So I go from the axis of revolution to this point Which is going to be capital R? But lo and behold, what's that distance from the x-axis up to that point on the curve? It's the y value on that curve. Well, that's the same curve Part of it's up here, and then it comes around and comes back down here. It's not a function Therefore, we don't have that kind of one-to-one thing going on. So when we talk about the y value For this particular location in terms of x we've got two different y values. That's an issue The first issue is solving this for why not delightful but possible The second issue is the inner radius and the outer radius They're just they're the same you describe them the same way So that tells me that the washer method is not going to work Okay, it's not going to work for two very good reasons So the only choice we have since these things are kind of washer looking is to go about Forming the volume a little differently besides chopping it up in this fashion perpendicular to the axis of revolution That works for the solid desk and it works for the washer But it doesn't work in this case because our washer method is failed. So we have to abandon that method so instead of I mean we're still going to form the three-dimensional solid the same way We're still going to go around the x-axis So it's still going to be a three-dimensional solid that has this Symmetric image down here when we look at it, but instead of chopping it up perpendicular To the axis of revolution. We're going to take elements of area little skinny little rectangles this time Parallel to the axis of revolution So we want rectangles That are parallel to the axis of revolution So very different in that regard from the solid desk method and the washer method tried washer it failed. So this is Possibly going to work for us. Now what we do with this little skinny little rectangle parallel to the axis of revolution is We spin it around revolve it around the x-axis and as we do so that forms a cylindrical shell It forms something that looks like this Does that help visualize what we're doing? There's some good diagrams in the book Yes, that entire page 456 Has a diagram actually the diagrams very similar to this one except it's oriented differently in the plane Based on how those cylindrical shells reform, but we have this one now I'm going to clutter up the diagram and I apologize for that, but The first little rectangular region that we're going to work with is very tiny right down here And we send that around and We form our first cylindrical shell right there. There's our kind of inside piece in this little children's toy Okay, there's the first one We continue to take these rectangles as we work our way out and each time we work our way out We form a cylindrical shell so we get a bunch of these concentric with one another Fitting right inside of each other and we continue to work our way out Not skipping like I'm skipping now. I'm trying to make it less cluttered But then we take this little region spin dinner spin it around the x-axis and we work our way out all the way to Here so here's our representative cylindrical shell. We know how much volume is in there 2 pi r h times the thickness What's the thickness going to be but how do we take this skinny little rectangle? It's actually labeled right the thickness is what? Isn't that it yeah delta y So the skinny little rectangle that we have It's thickness is delta y Which in the integrand for us is going to be dy That tells us we want to start at a certain y value What's the y value where we're going to start here? Zero right on the other side of zero so our first Little skinny little rectangle that we send around the axis of revolution is way down here Not much volume in that little cylindrical shell, but we start down here With these and we work our way up incrementally with little delta y's until we get to the y value what? To right so our limits on this problem are from y equals zero to y equals two That's where we take our skinny little rectangles our first one is right on the other side of zero our last one is right underneath here We take those skinny little rectangles send them around the x-axis. That's what generates the cylindrical shells So on this problem I'll bring the diagram back up as we need it. What have we determined thus far? Well, we determined the thickness Was a delta y or dy? Since we're integrating with respect to y we should expect to start at a certain y value. We decided that So we're starting with the skinny little rectangular Elements that we're going to spend around the axis of revolution right on the other side of y equals zero we continue to Use them till we get all the way up to y equals two To pi we can bring out front because there's nothing variable about that. So all we need is the radius and The height all right, so I'm going to bring this diagram back this distance From here to here and again, don't worry about the fact that the inner radius and the outer radius might be different We want them to be so thin that there's virtually no difference between the inner and outer radius So this is the radius of this cylindrical shell How would you describe what that radius is? We went from the x-axis up to a point here. It is right here. There's the radius of This particular cylindrical shell, how would you say what that is? That's the y value right because we went from the x-axis up So we would say that the radius is y So the radius is Y I'll go ahead and write it down. Am I going to have to change that or am I going to be able to leave that alone? Leave it alone. Why? Because we've got dy so the fact that the radius of this cylindrical shell is y and we're integrating with respect to y We're just going to leave it alone Okay, now we need the height Here's the height This distance from the x-axis, excuse me from the y-axis Over to the point on the curve here. It is right here. So that's the height What do we typically call the distance from the y-axis over to a point on a curve? We call that x. Is that right? From the y-axis over to this point ought to be the x value of this point So we come back here. What is the height? The height is x. Are we going to be able to leave that? No, we're going to have to change that and say well, that's fine. It is x But what is x equal to on this curve? Okay, and thankfully we know that we don't have to solve for that. So the radius That is described on each cylindrical shell as the y value of That particular point on the curve. The height is described as the x value the x value on this curve Based on the equation that was given to us. That is the x value And we would integrate with respect to y. Again when you get to this point on the problem You're a good two-thirds to three-fourths of the way home because this shouldn't be the difficult part at all Any question about any part of this Nicole? Is the r always going to be y and the h always going to be x? Anybody want to answer that? Okay, unless we're oriented the other way. So if you want to see an example Look at page 456 in your book and you'll see the cylindrical shells oriented differently They would be like this So if this if our cylindrical shells looked like this, here's our radius What would that be from the y-axis over to this point is x? That was not the case on ours, right and the height of this Would be from the x-axis up to this point and from the x-axis up would be Why so it does depend on how they're oriented in the plane? Oh, I know the problem I we finished a problem and I said we were gonna Go around a line that was parallel to one of the axes that's what I was going to start class with that I guess I won't be able to do that since the start of class already passed by But that's the type of problem we need to clean up before we leave this section So you kind of answered your own question there kind of depends on how they're oriented in the plane I'm not going to take this any further unless somebody has a question about how we get to a solution from here Everybody feel confident with that distribute the y Integrate each piece individually To pi is out part of the answer. We'll just save it to the end evaluate it from zero to two And it should be the number of cubic units bounded between Not between by revolving that curve around the x-axis Everybody okay with that to leave it and this could be a test question actually It is possible that I could say set it up only just to save some time Do not integrate nor evaluate. I've done that on a test before did I do that in the one 41? I May have on the exam just to save some time but once you get to this point Since integration was really at the end of 141. I'm kind of trusting that everybody has a pretty good mastery of this That a good good way to think that I trust that everybody has good mastery of that Okay, let's go back to that problem That I thought of while we were doing that one And it was a The first time through it was a washer problem And I said I wanted us to do that again. There we go Getting there Okay, there we go. So here's the problem we did yesterday and then I want to slightly Adapt it So we had y equals x squared plus two the equation was actually given to us a little differently than that But it's that equation. We also had a line y equals one half x Plus one and x equals zero x equals one now the problem we did We took this bounded region and we revolved it around the x-axis And fx plus one like this So we have that bounded region. There's the part of the parabola y equals x squared plus two Here's the line with y intercept one and slope of one half x equals zero and x equals one So we took that region we revolved it around the x-axis and we talked about the washers that resulted The other problem Let's say instead of revolving this around the x-axis. Let's revolve it around a line that's parallel to the x-axis Let's revolve this around the line Y equals three and how will that change? First of all what the region itself looks like or this three-dimensional solid what it's going to look like and then how's that going to change? As far as the equations that we set up and then ultimately the integration that we do I Think I know it helps me to draw this even though my drawings aren't The best in America Probably not even the best in this room There's the symmetric image on the other side of the line y equals three So we've got this three-dimensional solid that is formed So let's chop through it with Perpendicular to the axis of revolution so our axis of revolution is y equals three so we're going to chop through it in this fashion So one of these regions It's going to look like a what a washer Now are we going to be able to do it with the washer method Kind of depends on if we're able to describe the inner and the outer radius somehow So we're missing this volume in here. Is that correct when we slice through it? So each one is a washer. So here's our axis of revolution So this is kind of like a little cylinder on its edge With the volume missing out of the inside. What is the thickness? What are we dealing with here a delta? We chop it up this way we get little delta Exes right some increment of x going from here to here So our height is going to be delta x which in the integrand is going to be represented by how about dx instead How about the inner radius? This is going to get cluttered in a hurry. So let's Try the inner radius first so this distance from the axis of revolution to here Now I just did that incorrectly Because I went from the axis up to here that's not really anything related to my original curve So I probably want to go from the axis of revolution To one of my original curves so I can get that same inner radius by coming down. Is that correct? This is one of my original curves So this distance right here from the axis of revolution to this point on the curve Got to get a little Creative here What do we know this distance is from this point on the curve? Down to the x-axis. What do we typically call that we call that y? We know this entire distance is what? Three from the x-axis up to the axis of revolution is three So if we know from here to here is why and the entire distance is three. What do we want? Three minus why does that sound good? So the inner radius Sorry the diagrams cluttered and we kind of just got started is three which is this entire distance minus y Why on what why on the parabola is that correct? Three minus the y value on the parabola So let's go from the axis of revolution now to the outer radius. Well, that would be from here to here Let's Okay, we do know the value from here to here is the y value on the line We do know this entire distance is three So isn't it going to be three minus the y value on the line? Sorry, this is so cluttered. So we've got our height We've got our outer radius three minus the y value on the line We've got our inner radius Three minus the y value on the parabola. I think we're in business. I think this method is going to work Since we're going to be integrating this with respect to x Where do we want to start and stop this particular problem? Where do we start to form these washers in terms of x and where do they stop? Zero to one. All right, let's see what it looks like. So the pie will bring out front zero to one Capital r is three minus the y value on the line What is the y value on the line? So we're going to square that right? Three minus the y value on the line. There's the y value on the line for little r the inner radius We want that to be the distance from here to here, which is three minus the y value on the parabola What is the y value on the parabola? So I guess the question I need to pose if I were Trying to teach this adequately which actually I am some of you might find that hard to believe with a cluttered-up diagram like that Were we able to describe the inner and outer radius? differently so that we can accommodate the fact that the inner radius and the outer radius are actually two different Values we were they look the same here But once we decided which y value the y value on the parabola y value on the line We are able to describe them distinctly so this method is going to work if you're not able to describe them distinctly It's not going to work. So we've got the pie. We took it out front. We've got our limits from zero to one outer radius three minus the y value on the line We have squared that Minus the inner radius squared three minus the y value on the parabola and our height is Some increment of x or some delta x or dx in the integrand I would recommend doing that arithmetic first before you square it And the same thing here So if we were to do one more step on the setup What are we going to be squaring? What is three minus that quantity? Two minus a half x that right in here three minus this quantity is really going to be what? One minus x squared, so we've got a little work to do before we can actually Integrate and evaluate, but I don't think anything. We're going to encounter is going to be Other than middle of the road kind of integration. We've done to this point time Square this remember you're going to get a middle term right which is twice their product Remember when you square this you're going to get a middle term which is twice their product Anybody we need to go any further on this? to get a solution so in in that way of a kind of an overly brief Summary if we're going to do if we've got a volume of a solid a revolution We could have a solid desk Which is going to be some version of that If they're not solid, but they're washers when you chop them up perpendicular to the axis of revolution I'll tell you the most common error. I've seen here is Big R minus little R the quantity squared That's not the way we were able to derive the volume in a washer Outer radius squared minus inner radius squared and then if neither one of those work and we have to jump to the cylindrical shell method 2 pi r h Establish what the thickness is of that Little rectangular region that we're spinning around the axis Let me see before we stop this will take about a minute, and then we'll wrap it up this same problem Actually, let's go all the way back to the first problem So let's go back to the one where we wrap this around the x-axis Now we did this problem using the washer method. Could we do this problem? using cylindrical shells Let's see Since this is our axis of revolution We would want to take little skinny little elements of area little rectangles Parallel are we going to be able to come up with a description for the radius of Each cylindrical shell that's going to work for all of them and The height of each cylindrical shell that's going to work for every one of these Some of you are shaking your heads now. Why not? What do you see a problem here? You're not all bound by the same okay, we've got different boundaries right for for these we had a curve before that Looked like this so when we described that piece and we were asking the question is that description going to work for this piece Wasn't it always from the y-axis over to this point? From the y-axis over to this point. We don't have that down here. Do we? Because here we're from the y-axis over to this point on the line and That's going to work for all these right till we get here And then we're going to have from the y-axis over to here, which is really just what one all the way up to here and Then we've got a little change when we go up here. What is this distance? Well, we're not even coming from the Y-axis over to here. I think we're going to ask that's going to be trouble You could break it up into pieces and do so a nickel shell But unless we can describe each Little element of area that we form parallel to the axis of revolution the same way this method is going to be a little tricky But we don't need it anyway to me that method is a little more difficult than the Washer method anyway, so if washer works we ought to use it if it doesn't we ought to see if So the nickel shell is going to work Okay, I think we are reasonably close to where we should be according to the syllabus As I told you before class. I'm going to adjust deadlines for web assign So when I'm done with that I'll send an email to you so you'll know that deadlines have changed. Have a great weekend