Washer method rotating around non-axis

Now let's do a really
interesting problem.

So I have y equals x,
and y is equal to x

squared minus 2x
right over here.

And we're going to
rotate the region

in between these two functions.

So that's this region
right over here.

And we're not going to rotate
it just around the x-axis,

we're going to rotate it around
the horizontal line y equals 4.

So we're going to
rotate it around this.

And if we do that, we'll get
a shape that looks like this.

I drew it ahead of time, just
so I could draw it nicely.

And as you can see, it looks
like some type of a vase

with a hole at the bottom.

And so what we're going to do
is attempt to do this using,

I guess you'd call it
the washer method which

is a variant of the disk method.

So let's construct a washer.

So let's look at a given x.

So let's say an x
right over here.

So let's say that we're
at an x right over there.

And what we're
going to do is we're

going to rotate this region.

We're going to give
it some depth, dx.

So that is dx.

We're going to rotate
this around the line y

is equal to 4.

So if you were to visualize it
over here, you have some depth.

And when you rotate it
around, the inner radius

is going to look like the
inner radius of our washer.

It's going to look
something like that.

And then the outer
radius of our washer

is going to contour
around x squared minus 2x.

So it's going to
look something--

my best attempt
to draw it-- it's

going to look
something like that.

And of course, our washer
is going to have some depth.

So let me draw the depth.

So it's going to
have some depth, dx.

So this is my best attempt at
drawing some of that the depth.

So this is the
depth of our washer.

And then just to make the face
of the washer a little bit

clearer, let me do it
in this green color.

So the face of the
washer is going

to be all of this business.

All of this business is going
to be the face of our washer.

So if we can figure
out the volume of one

of these washers for a
given x, then we just

have to sum up all of
the washers for all

of the x's in our interval.

So let's see if we can
set up the integral,

and maybe in the
next video we'll

just forge ahead and actually
evaluate the integral.

So let's think about the
volume of the washer.

To think about the
volume of the washer,

we really just have to
think about the area

of the face of the washer.

So area of "face"--
put face in quotes--

is going to be equal to what?

Well, it would be the
area of the washer--

if it wasn't a washer,
if it was just a coin--

and then subtract out
the area of the part

that you're cutting out.

So the area of the
washer if it didn't

have a hole in the
middle would just

be pi times the
outer radius squared.

It would be pi times
this radius squared,

that we could call
the outer radius.

And since it's a washer,
we need to subtract out

the area of this inner circle.

So minus pi times
inner radius squared.

So we really just
have to figure out

what the outer and inner radius,
or radii I should say, are.

So let's think about it.

So our outer radius is
going to be equal to what?

Well, we can visualize
it over here.

This is our outer
radius, which is also

going to be equal to
that right over there.

So that's the distance
between y equals

4 and the function that's
defining our outside.

So this is essentially,
this height right over here,

is going to be equal to 4
minus x squared minus 2x.

I'm just finding the distance
or the height between these two

functions.

So the outer radius is
going to be 4 minus this,

minus x squared minus
2x, which is just 4

minus x squared plus 2x.

Now, what is the inner radius?

What is that going to be?

Well, that's just going to
be this distance between y

equals 4 and y equals x.

So it's just going
to be 4 minus x.

So if we wanted to find
the area of the face of one

of these washers for a
given x, it's going to be--

and we can factor
out this pi-- it's

going to be pi times the
outer radius squared,

which is all of this
business squared.

So it's going to be 4 minus
x squared plus 2x squared

minus pi times
the inner radius--

although we factored
out the pi-- so

minus the inner radius squared.

So minus 4 minus x squared.

So this will give us
the area of the surface

or the face of one
of these washers.

If we want the volume of
one of those watchers,

we then just have to
multiply times the depth, dx.

And then if we want to actually
find the volume of this entire

figure, then we just have to
sum up all of these washers

for each of our x's.

So let's do that.

So we're going to sum
up the washers for each

of our x's and take the
limit as they approach zero,

but we have to make sure
we got our interval right.

So what are these-- we care
about the entire region

between the points
where they intersect.

So let's make sure
we get our interval.

So to figure out our
interval, we just

say when does y equal
x intersect y equal

x squared minus 2x?

Let me do this in
a different color.

We just have to
think about when does

x equal x squared minus 2x.

When are our two functions
equal to each other?

Which is equivalent
to-- if we just

subtract x from
both sides, we get

when does x squared
minus 3x equal 0.

We can factor out an x
on the right hand side.

So this is going to be when does
x times x minus 3 equal zero.

Well, if the product is equal
to 0, at least one of these

need to be equal to 0.

So x could be equal to 0,
or x minus 3 is equal to 0.

So x is equal to 0
or x is equal to 3.

So this is x is 0, and
this right over here

is x is equal to 3.

So that gives us our interval.

We're going to go
from x equals 0

to x equals 3 to get our volume.

In the next video,
we'll actually

evaluate this integral.