[MUSIC].
We can cut a three dimensional object up into shapes beyond just slabs, disks,
washers. Specifically instead of building an
object out of disks, instead of building it by stacking up disks of varying radii.
We can build the object by wrapping it with these shells.
This shell thing is supposed to be like a cardboard mailing tube.
It's a cylinder sort of a thin wall. But how are shells different from
washers? Well there are some similarities.
Both shells and washers are things that you get by taking cylinders and drilling
out other cylinders. But for washers, the thin dimension is
its height. And for shells, the thin dimension is
sort of the thickness of this wall. We need to write down a formula for the
volume of a shell in terms of it's height, it's radius and that thin dr.
Let me label those parts on the diagram here.
So the height of the shell. I'll call h, the thickness of the shell.
I'll call that dr, and the radius of the shell I'll just call r.
Now, if I get out my scissors of mathematics here and cut the shell.
You might imagine unrolling the shell, and then you just end up with a slab like
this. How big is this slab?
Well, the slab's height is still h. The thickness of this slab is dr.
And how long is this slab? Well the slab's as long as the
circumference of the circle , which is 2 pi r.
And multiplying these three dimensions will give me the volume of the slab.
So you're guessing that the volume of the shell or the slab is 2 pi rh dr.
We can say something a bit more precise. Well here's the shell again.
But instead of calling this dr. Let's think of this as a big cylinder,
with a little cylinder drilled out. So both the cylinders have the same
heighth, h. But let's say the big cylinder has a
radius, r. And the little cylinder has a radius
little r. Let's write down the formula for the
volume of the shell in between. So now the volume of the shell, right, is
the volume of the big cylinder pi big R squared h.
Minus the volume of the little cylinder that I drilled out, pi little r squared
h. In effect you'll recognize that this,
this makes it look a lot like a washer. But here's the difference, little r is
really close to big R. Right, I'm thinking of their difference
as dr. Right, very thin walled cylinder.
Well lets rewrite this volume anyhow. I'm going to factor this out pi.
Big R squared, minus little r squared, h. Now let's rewrite this.
This is a difference of squares, so I can write that as pie, big R plus little r.
Big R minus little r, h. And thinking of the difference in these
radii as being really small. So I'm going to rewrite this term as dr.
So this is pi, big R plus little r, and I'll write h dr.
Now what do I do with big R plus little r?
Well this is the sum of these two radia. I mean, if you think about big R and
little r as being really close together. I might as well just write this as 2
little r. So I'll write this 2 pi little r.
That's 2 little r's, if I'm imagining big R and little r to be close together, h
dr. And you'll see this is the same as the
formula that I got before for the volume of a shell.
This formula, the formula for the volume of a shell, to write down an integral
that calculates the volume of a solid of revolution.
As an example, let's take this region. This is the graph y equals 1 minus x
squared. And I just want to consider y greater and
equal to 0, so consider this region inside here.
So let me write y less than or equal to 1 minus x.
So just to emphasize this, it's this region inside here, and let's take that,
and let's rotate that region around the y axis.
To give myself this solid of revolution, I'm going to calculate the volume of that
solid. Admittedly, we could do this problem
without using shells, but using washers. In that case, I'd be chopping the region
up into horizontal slices. And each of these horizontal slices, when
I rotate it around the y axis would give me a disc.
And then I just integrate the volumes of those discs to give me the volume of the
solid of revolution. But let's try it with shells.
In this case, I'll cut it up into vertical slices and I'll rotate these
vertical slices around the y axes to produce shells.
And I'll integrate the volumes of those shells.
To give me the volume of the whole solid of revolution.
Now, we're going to use the formula, for the volume of a shell.
Let's just look at one of those shells. So, here's one of these thin rectangles,
I imagine, rotating this thin rectangle, around.
The the y axis to produce this shell. Now, how how tall is this shell?
Well, the height of that shell is given right here.
It's 1 minus x squared. The radius of the shell is x and the
thickness of that shell is dx. So, I put all those pieces together.
I can write down the volume of just that one, one shell.
And it'll be 2 pi, the radius of that shell.
The height of that shell, the thickness of that shell.
Now I have to worry about the end points of integration.
You might think that I'd be integrating from x equals minus 1 to 1.
But actually I only have to integrate from x equals 0 to 1.
Because rotating a thin rectangle over here also accounts for stuff happening
over here. So it's enough just to integrate from x
goes from 0 to 1. So I'll write that here, x goes from 0 to
1. This integral will calculate the volume
of my solid of revolution. Finally, it's just a matter of evaluating
the integral with the fundamental theorem of calculus.
And to do this integral. Well I'll pull out these constants, 2 pi,
the integral from 0 to 1 of x minus x cubed, when I combine these two terms.
Now I just have to write down an antiderivative here.
2 pi, antiderivative of x is x squared over 2, anti-derivative x cubed is x to
the 4th over 4. And I'm evaluating this at x equals, 1,
and x equals 0. And I'm subtracting, alright?
That's the fundamental theorem of calculus.
But when I plug in 0, I just get 0. So the answer is just whatever I get when
I plug in 1. And, that's 2 pi, a half minus a 4th, and
a half minus a 4th is a 4th. So, 2 pi times a 4th, and 2 times a 4th
is a half. So, pi over 2, is the volume of my solid
of revolution. We did it.