We've been doing a lot of rotating around the x-axis, so
let's start rotating around the y-axis and see what we can do.
Or at least attempt to.
Let's me draw my axes.
That's y-axis.
That's my x-axis.
Well let's just do it with an example, but we'll call it f
of x too because it'll be generalizable.
Let's just draw y equals x squared.
Let me just draw the positive because we're going to rotate
it around the y-axis and it's symmetric anyway, so that's
y equals x squared.
This is y-axis.
This is x-axis.
Actually, no I'm going to keep it general, then we'll actually
solve it particularly.
So we'll call this f of x, but clearly this is
y equals x squared.
This is f of x.
And we know how to take the volume if I were to rotate
this around the x-axis.
But what if I wanted to say-- I guess we could call it the area
between 0 and-- I'm trying to determine how general to be.
Well let's just say between 0 and 1.
I think the boundaries might make sense to you.
Roughly this area, and I'm going to rotate it
around the y-axis now.
So what's that final figure going to look like?
The base of it-- let me see how well I can draw it.
Nope that's not what I wanted to do.
The base is going to look something like a
cylinder like that.
And then the top of it is also going to be-- no, that's
not what I wanted to do.
Let me draw the side lines.
So it's going to look something like that.
And then the top of it looks something like that.
But it's not just going to be cylinder, right?
If I was doing this entire block it would be a cylinder.
But the inside of it is going to be kind of hollowed out.
Let me see how effective I am at drawing that.
I'll do it in a different color.
So the inside is going to be hollowed out.
I don't know if that makes sense to you that.
It's kind of like on the inside it'll look like a bowl.
On the outside it'll look like a cylinder or a can.
Hopefully that makes sense.
You take this and you rotate this around.
And the curve that specifies the inside would be y
is equal to x squared.
It would rotate all the way around.
I think that makes sense.
The drawing is the hardest part.
So how do we do it?
Well even the shape might give you an idea.
We can't use that disk method, what we were doing before when
we were rotating the x-axis, that was the disk method,
because we were essentially imagining each of these
particular disks and then summing them up.
Now we're going to do something called the shell method.
So what's the shell method?
Instead of taking a bunch of disks and figuring out their
combined volumes, we're going to take a bunch of shells.
So what's a shell?
So imagine a rectangle right here.
Hope you can see it right there.
Let's say it's at the point x,1.
What's its height going to be?
Its height going to be f of x,1.
That's its height.
Now imagine taking that sliver and rotating
it around the y-axis.
What's it going to look like?
Well, it's going to look like a shell, it's going to look like
a cylinder, just like the outside of a cylinder.
It's going to look not too different then that but I want
to draw it well because intuition is the most important
thing that, not getting the problem right.
Let me see if I can draw this respectably.
And then we're going to have the bottom of the shell, it'll
look something like that.
Let me finish these lines up.
I think you get the point.
OK.
So it's going to look like a shell like that.
The outside of the shell is going to be solid.
And it'll have some width, but the inside is hollow.
Let me do a different color.
Maybe a darker color to show that that's the inside.
You know it's like a ring essentially.
And so what's the height of this ring?
The height is going to be f of x,1.
So let me do a brighter color so you know what I'm saying.
The height of this ring is f of x,1.
f of x evaluated at that arbitrary point we picked up.
What is going to be the surface area of this ring?
You know, this outside.
Well let's think about it.
It'll be the circumference of this ring times it's height.
So what's the circumference of this ring?
Let's go back to our basic geometry.
Circumference is equal to 2 pi times the radius.
So if we know the radius of it, we know the circumference.
Well what's the radius?
Well the radius is how far we went from the axis of
rotation to that point.
So that's the radius.
So in our particular example the radius is x,1.
It's that x point that we're evaluating it at.
So circumference is going to be equal to 2 pi times that point
that we're evaluating at.
And so the surface area-- this magenta thing that I filled
in-- that's going to be equal to the circumference times this
height, which we already said is f of x,1.
Let's call it area surface.
Surface area is equal to circumference times height,
which is equal to 2 pi x,1 times f of x,1.
We figured out the surface are of this.
Now how do we figure out the volume?
Well what's the width of it?
How thick is this ring?
What's this thickness right here?
It's a very small thickness.
But we took this sliver, and this sliver as we learned in
previous calculus, the width of this little rectangle is dx.
And you know when we take the integral, it's going to get
infinitely smaller and smaller and we'll have infinitely
more and more of them.
So the width of this is dx.
Let me draw it big, not so horrible looking.
So if this is a sliver, it's width is dx.
It's height is f of x,1.
x,1 will be right in the center.
And then it's distance from the center is of course x,1.
Hopefully that make sense.
So what's the volume of this shell?
So the volume of the shell-- this shell, not this one-- the
volume of the shell is going to be equal to the surface area
of the shell times how wide that surface is.
And that width is dx, so it's going to equal this times dx.
So the volume of that shell is 2 pi x,1 times
f of x,1 times dx.
I think you see where I'm going with this now.
So what would be the volume of the entire rotated
figure, this thing here?
Well I'm just going to sum up each of these shells.
I have one shell there, then here I'll have a slightly less
high shell, and up here I would have a much bigger shell,
and I'll add them up.
Here's one shell that goes around.
Then they'll be another shell here, and I'll add them all up.
And that's taking the integral.
So the total volume of the figure when I rotate it around
the y-axis is going to be-- and my boundary is from 0 to 1-- 2
pi-- this one I just told you a particular x,1 but we're going
to sum them over all of the x's.
So it's going to be 2 pi x f of x dx.
This is just a constant, so you could call it
2 pi times x f of x.
So let's take a particular example.
Let's do it for x squared.
Let's say the function is x squared.
So in this case the volume is going to equal-- let's take the
2 pi out-- 2 pi integral 0 to 1 x times f of x-- f of x in our
case is x squared, which I drew earlier-- dx equals 2 pi.
This is just x to the third, right?
x to the third.
So it's going to be 2 pi times the antiderivative
of x to the third.
Well that's x to the fourth over 4.
Evaluate it at 1 minus evaluate it at 0.
Well that equals 2 pi times 1 to the fourth is 1, so
1/4 and then minus 0.
So it's 2 pi times 1/4.
So that's pi over 2.
That's the volume, and we just rotated it around the y-axis.
I will see you in the next video.