WEBVTT

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Now let's do a really
interesting problem.

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

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squared minus 2x
right over here.

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And we're going to
rotate the region

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in between these two functions.

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So that's this region
right over here.

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

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

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So we're going to
rotate it around this.

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

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

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

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with a hole at the bottom.

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

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I guess you'd call it
the washer method which

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is a variant of the disk method.

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So let's construct a washer.

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So let's look at a given x.

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So let's say an x
right over here.

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

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

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going to rotate this region.

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

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So that is dx.

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We're going to rotate
this around the line y

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is equal to 4.

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

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And when you rotate it
around, the inner radius

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

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It's going to look
something like that.

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And then the outer
radius of our washer

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is going to contour
around x squared minus 2x.

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So it's going to
look something--

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my best attempt
to draw it-- it's

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going to look
something like that.

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

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So let me draw the depth.

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

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

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So this is the
depth of our washer.

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

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clearer, let me do it
in this green color.

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So the face of the
washer is going

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to be all of this business.

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

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So if we can figure
out the volume of one

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of these washers for a
given x, then we just

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have to sum up all of
the washers for all

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of the x's in our interval.

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

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and maybe in the
next video we'll

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just forge ahead and actually
evaluate the integral.

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

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To think about the
volume of the washer,

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we really just have to
think about the area

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of the face of the washer.

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

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is going to be equal to what?

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

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

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and then subtract out
the area of the part

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that you're cutting out.

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So the area of the
washer if it didn't

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have a hole in the
middle would just

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be pi times the
outer radius squared.

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It would be pi times
this radius squared,

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that we could call
the outer radius.

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

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the area of this inner circle.

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So minus pi times
inner radius squared.

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So we really just
have to figure out

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

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So let's think about it.

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

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Well, we can visualize
it over here.

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This is our outer
radius, which is also

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going to be equal to
that right over there.

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So that's the distance
between y equals

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4 and the function that's
defining our outside.

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So this is essentially,
this height right over here,

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

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

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functions.

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

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minus x squared minus
2x, which is just 4

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minus x squared plus 2x.

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Now, what is the inner radius?

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What is that going to be?

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

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equals 4 and y equals x.

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

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

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

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

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going to be pi times the
outer radius squared,

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which is all of this
business squared.

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

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minus pi times
the inner radius--

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although we factored
out the pi-- so

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minus the inner radius squared.

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So minus 4 minus x squared.

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So this will give us
the area of the surface

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or the face of one
of these washers.

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If we want the volume of
one of those watchers,

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

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

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

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for each of our x's.

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So let's do that.

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

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

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

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

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between the points
where they intersect.

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So let's make sure
we get our interval.

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So to figure out our
interval, we just

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say when does y equal
x intersect y equal

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x squared minus 2x?

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Let me do this in
a different color.

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We just have to
think about when does

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x equal x squared minus 2x.

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When are our two functions
equal to each other?

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Which is equivalent
to-- if we just

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subtract x from
both sides, we get

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when does x squared
minus 3x equal 0.

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

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

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

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need to be equal to 0.

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

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

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

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is x is equal to 3.

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So that gives us our interval.

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We're going to go
from x equals 0

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to x equals 3 to get our volume.

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In the next video,
we'll actually

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evaluate this integral.