Calculus: Shell Method-Integrating w Respect to y

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Applications of Integral Calculus
Shells
The Shell Method - Integrating with Respect to y Page [1 of 3]
All right, let's see if we can find the volume of this particular solid of revolution. And first, what we do is take a look at the following region bounded by these curves: y = , the y-axis, y = 1, y = 4. And I want to revolve the whole thing, now, around the x-axis. So what would this thing look like? First I'll just graph this region in the plane before I revolve, do something to revolving.
So what do I have here? I've got y = . What's the graph of that look like? Actually, that is half of a wing of a parabola on its side, and you can see that if you were to square both sides. If you square both sides, you see that x = y^2. And x = y^2 is just a happy-faced parabola looking like this. But since I only have the positive square root, I only pick up this part of the wing. So in fact, it's just this parabola just like that. And then I've got the y-axis and I've got y = 1 and y = 4. So you're going to see I've been squashing a lot of stuff here. In fact, if I had any brains at all, I would have had this be a little bit taller, but that's okay. You don't even need brains to do this question, as I will demonstrate for you live.
So let's put in the parabola right now - this fabulous parabola. So we have this thing. I'm putting the parabola right now. Just start to see, there's no down wing, because it's just all up wing. So there's the parabola, y = . And the y-axis - that's just here. Okay, fine. And y = 1. And remember, this is just for us, so in fact, let me deliberately, for pedagogical purposes, make the scale all messed up. So here's 1 and here's 4. Doesn't make a difference. It's just for us to get a sense of what's going on. There's the region.
And so that region looks like this - looks like a trapezoid that's been pushed in like this. Now I take that region and revolve it now around the x-axis. So now imagine hinging up the entire world and just flipping it around and you continue to flip it all the way around. Well what would it look like? I'd have this reflection down here, so let's draw this the best we can, as quickly as we can. We have this. We have this goes to here.
By the way, I encourage you to try to do things quickly. Sometimes there's great value in practicing to do something really fast. Accuracy is extremely important, of course, but it's also worthwhile, again, the habit of trying to do things as quickly as you can, just because if you have to take a test or something, sometimes they don't give you all the time in the world. They only give you limited time. So it's good to practice to do things quickly. And if you don't do it right, that's okay. You can go back and do it right later. Of course, on the test, you can't always do that unless you're allowed to take the test again. That would be great. You can't always do that though.
Okay, so that's the region. It sort of looks like a cylinder that has a very, very narrow hole in it, so it has thick walls. It's like an enforced cylinder. And then someone sort of made the inside of it - sort of rounded it out a little bit. So it's rounded out here and then the cylinder sort of takes the rest of the shape down there. So what I want to do is figure out what the volume of that is. So how would I do that? Well, if you wanted to do it by washers, you could do that by washers by slicing this way, but notice that you come to a point where the washer shape will change. If I look from here to here, the outer diameter will be this and the inner diameter will be this little line. But the moment you cross that interface, then the washers take on a different shape. They go from this outer diameter now to this diameter formed by the curve. So you'd have to set this up as two separate integrals and then figure out how all the things fit together.
A slightly better method is to realize that this is a great opportunity to use the shell method. Because really, if I stack things like this, I see that, in fact, I could put a whole bunch of shells in there. And if I do some coring out, I'm going to core this way now, and if I core concentrically, concentric coring will lead to a thing like this. And that's just the object, if you think about it, on its side.
So I'm going to core things out and so how am I going to set up the integral? And again, this is always the most challenging part and this is what we're really after here is the understanding of how to set this integral up. Let's put in an arbitrary core piece. This is how I always operate. I put in a random one. So I put one in at a random height here. And so in fact, let's pretend I'm at an arbitrary height of y. And so here is the little core, and that's an arbitrary height of y. And there is one of my shells, just like that, right in there. You can see the shell part here, but it's fitting in just like that.
Now what I want to do is I want to sum that up. Now where do I sum them up from? From the narrowest waist size to the widest waist size. And waist size here is really just the radius. So it's basically going from the smallest radius to the largest radius, but just think of it as a waist size. What's the tightest one and then what's the widest one?
So I'm going to integrate, and so the tightest waist size is going to be when y = 1. So I go from 1, and then the waist size goes all the way up to 4. And now what am I going to do? I'm going to actually find the volume of one of these slices - one of these cored pieces. So what is that? If you unfold it, you just get a rectangular piece like this.
So now what's the area of that outer surface, and thus figuring out the volume of the shell? Let's see, if we unfold it, it would look like this. By the way, this is not a crazy idea to sort of actually do this when you do these questions. And you cut, cut, cut open and it comes out like this.
So the radius here - that radius is what? Well, that's from the very center out to where we are. That's y. So the radius here is y. So the radius equals y. And so what's this length? Well that length is just the circumference of that circle. So that would be 2r, which in this case, is y. So 2y is this length. What's this length right here? Well, that length is the length from here out to here. Now how do I find that? The question is how far is this away? So what's the x value here associated with that? How would you figure that out? You've got to come back her and say, what is this x value? If I'm at y, what does x equal?
If you look back at this thing, you can figure out what x equals. If you solve this for x, x would have to equal y^2. So in fact, this x value - that's the base here - that must be y^2, because if I'm at a height of y, the place where that y hits this curve - that happens at an offset of x^2. If you don't believe me, check. Put in x^2 right in there. I have , and that just gives me x. And so indeed, that's exactly what I want to hit.
I'm sorry, what am I saying? If you put in y^2 for x, you see the . is y, and that's what I wanted. I wanted to get y. So what I want to do to find out what this x value is, I come back here and solve this for x. Square both sides. x = y^2. So that's y^2.
And then what's this little thickness? Well, that little thickness is a little teeny change like this. It's a change in which direction? It's a change in the y direction. So this must be a dy - that's the thickness. And so what's the volume of this particular shell? It's going to be 2y(y^2)dy. And so that's just 2y^3dy. And as you can see, as always, the integral itself is a piece of cake. The real challenging part is to get that integral.
So let me review this thing. Since I'm coring around the x-axis, that means that this is going to be a y integral, because the waistline is going to be a y value. Since I'm coring this way, the tightest waist will be a y = 1, out to the largest waist, y = 4. So it's going to be a y integral. And then the shells themselves, if you draw them in there, well the radius from out to here is the y value. So there's that y. And then so if I want to find the length around, that's 2y. And then what about the height? The height is just the x value when you're at that y. So you go back to here and I solve this for x and I see a y^2. And so in fact, that's the y^2 length there. And the thickness is a tiny change in the up and down direction, so that's the change in y, and I see this.
So now what you've got to set up, of course, this is now pretty straightforward, because you just take 2, and that equals 2. And I evaluate this from 1 to 4. So when I plug in 4, I see 2, and I subtract 2, and if you do that, I think you see 128 - , which equals 255, if you get a common denominator, all over 2. And so the volume of this particular region, described in this rotational way, turns out to be units cubed.
The important thing, I think, is to realize how to go from this picture to this integral. The trick really is to think of these things as shells and think of the waist size. And the waist size provides us with a way of figuring out the endpoints. And then if you unfold one of the shells and open it up, you can slowly and carefully figure out what the volume of that particular shell is. Putting that all together, what you see is the integral, which happily can be done with very little effort, and now you've got the volume using shells. I'll see you at the next lecture.