Welcome to a video
on The Washer Method
for Determining Volume.
This video will show
rotation about the y-axis.
And again, the washer
method is very similar
to the disk method.
You could think of a
washer as a hollow disk.
So, just to review,
remember the disk method is based upon
determining the volume of cylinders,
which can also be called disks,
where the volume is
equal to pi r squared h,
where r would be the radius,
and this would be the height.
And if the rotation was about the y-axis,
this would be equal to delta y.
For a washer, we'll have to
determine the volume
of the outer cylinder,
and subtract the volume
of the inner cylinder.
So, this would be big R,
and this would be little r.
And again, this would the height,
and with this orientation,
would be equal to delta y.
So, we'll combine this
idea with integration
to determine the volume of a solid.
So, let's take a look at
how we're going to do that.
If we start with an area or
region bounded by two functions,
as we see here, and then
rotate about the y-axis,
you would produce a solid
that looks like this.
And the goal is to
determine the volume of this
using the washer method.
And that would look something like this.
If we have this solid,
rotated about the y-axis,
and try to determine the
volume using very thin washers,
here's a sample of eight
different washers we could use.
And as the number of
washers approaches infinity,
the accumulation of these volumes
would equal the volume of the solid.
So, the key to setting up
this integral that will give us the volume
is to consider a representative rectangle,
bounded by these two functions
that would represent one washer.
So, to do that, we have
to make a rectangle that is perpendicular
to the axis of rotation.
So, it would look something like this.
And now, since the width or height
of this rectangle is delta y,
we will have to integrate
with respects to y.
If you take a close look
at this integral formula,
you can see the volume
of a cylinder in here,
where we have pi r squared,
and then delta y would
represent the heights.
So, the outer radius is big R of y,
and that will be the
distance from here to here.
On this distance is x,
which must be expressed in terms of y.
So, we call this R of
y, for the outer radius.
And the inner radius would be the distance
from here to here.
Again, that distance is equal to x,
but it must be expressed in terms of y.
So, we call this little r of y.
And then lastly, we integrate
on the interval from c to d,
along the y-axis.
So, this would be c,
and based upon this
intersection point here,
this would be d.
So, when we use the washer method,
one of the most important
things to remember
is that the representative
rectangle here in black
will be perpendicular
to the axis of rotation.
Let's do an example.
We want to determine
the volume of the solid
generated by the bounded
region of the given equation.
So, we have y equals
negative x squared plus four,
that's the blue function.
And we have x equals one,
the vertical line in red.
And then we have y equals zero.
So, our region that we're going to rotate
will be this region here.
And again, we're rotating
about the y-axis.
So, we're rotating around here.
And if we did that,
we'd get a solid that
looks just like this,
where it looks like a paraboloid,
where the top was cut
off and it's also hollow.
So, we'll start by sketching
a representative rectangle
that's going to be perpendicular
to the axis of rotation.
So, it'll look something like this.
And the height of this
rectangle is delta y,
and that's why we have to
integrate with respects to y.
So, let's go ahead and
see if we can set this up.
We have the volume is equal to pi,
times the integral from c to d,
which would be from here to here,
which is from zero to three.
Next, the outer radius
is determined by the
distance from the y-axis
to the blue function.
That would be this distance here.
This distance is equal to x,
but it must be expressed in terms of y.
And it's determined by the function
y equals negative x squared plus four.
So, we actually have to take
y equals negative x squared
plus four and solve it for x
because this distance here is equal to x.
So, let's go ahead and move the
x squared to the other side,
and then also subtract y on both sides.
And if we square root both sides,
we have x equals the square
root of four minus y.
And we're in the first quadrant here so,
so we're not going to
have a plus or minus here.
So, big R of y is equal to the
square root of four minus y.
And luckily, this is squared,
so that's going to undo the square root,
minus little r of y.
Well, little r of y is this distance
from the y-axis to the vertical line,
and this is always going to equal one.
So, the inner radius is just equal to one,
and then we have it squared,
and we integrate now with respects to y.
Let's go ahead and square these.
So, we have pi, this
should be four minus y,
minus one,
with respects to y.
Let's go ahead and clean
this up a little bit.
Looks like we'll just have three minus y.
Let's go ahead and find
the antiderivative here.
So, we have three y,
minus y to the second divided by two,
and we'll evaluate this at three and zero.
So, first we'll replace y with three.
Then replace y with zero.
And both of these terms
would be equal to zero.
This comes out to nine pi over two,
which would be the volume
of this solid that we see here.
Okay, the next video we'll talk about
how to use the washer method
when rotating about axes
other than the x- and y-axis.
Thank you for watching.
