Welcome to the first of several videos
on the shell method
for determining volume.
This video will only show
rotation about the y-axis,
there'll be other videos that show
rotation about other axes.
So, we've already talked about
how to determine the volume
of a solid using the
disc and washer method.
Let's take a look at
an animation to develop
the idea of how we're going
to use cylindrical shells
to determine the volume of a solid.
So, if we consider this shaded region here
rotated about the y-axis, we know by now
that it would produce a solid.
But if we take a look at just the solid
created by rotating these
very thin red rectangles
about the y-axis, it would
produce something called
the cylindrical shell,
as we see here in red.
And the idea behind the shell method is
as we increase the number of shells,
the volume will approach
the volume of the region
rotated about the y-axis, as we see here.
Now, if we go back and take a look
at just one of the shells,
we have to ask ourselves,
how would we find the
volume of just one shell?
Well, we'd have to find the
surface area of the shell
and multiply it by its thickness.
Let's take a closer
look at just one shell.
If we wanted to rotate the region bounded
by the x- and y-axis
in this blue function,
we know it would produce a solid
that looks something like this.
And this very thin, hollow
cylinder would represent
one shell that we could use
to determine that volume.
One thing we should be
able to recognize is
if we cut this cylindrical
shell and stretch it out,
it would be a long rectangle.
And this distance here would
come from the circumference
of this circle, which
is equal to two pi r,
and this distance here
would be the height.
So the surface area of
the cylindrical shell,
so the surface area of the shell
would be equal to two pi r times h.
And then to determine
the volume of the shell,
we'd have to multiply it by its thickness.
Well, the thickness of the shell would be
the thickness of this very thin rectangle
that was used to create the shell
as it rotated about the y-axis.
And this width would be equal to delta x.
So to find the volume of the shell,
we have the volume equal to
two pi r h times the width,
and all this will be equal to
two pi times what we'll call
r of x, the radius function, times h of x,
the height function, and
the width will be delta x.
So the volume of the solid would be equal
to the sum of the volumes of the shells
as the number of shells approach infinity.
So to formalize this
using the shell method,
with a vertical axis
of rotation the volume
will be equal to this definite integral,
where r of x is the radius,
h of x would be the height,
and the limits of integration, a and b,
are the interval on the x-axis.
In order to set this up,
it's important to sketch
a representative rectangle
just like we did in the washer method.
However, for the shell method,
a representative rectangle
will be parallel to the axis of rotation.
So if we're rotating about the y-axis,
we could use this as a
representative rectangle.
And the width of this rectangle
would be the width of the shell,
this would be delta x,
and that's a reminder
that we'll integrate with respects to x.
r of x would be the distance
between the axis of revolution
and the center of the rectangle,
so this would be our r of x,
and then h of x is the
height of the rectangle,
and that would be this distance here.
And then lastly, we integrate from a to b,
where a and b is the
interval along the x-axis,
so this would be a and this would be b.
And just to emphasize this one more time
let's go ahead and take a
look at one more animation.
If we wanted to use the method
of cylindrical shells
to determine the volume,
so to find the volume
using shells we'd start
with one shell and then
start accumulating the volume
of the individual shells,
and as the number of shells
approached infinity, we would approach
the volume of the actual solid.
Okay, let's go ahead
and look at an example.
We want to use the shell
method to determine
the area bounded by the region
rotated about the y-axis.
So, since we're rotating about the y-axis
we will sketch a rectangle
that would represent one shell,
and since we're rotating about the y-axis,
our representative rectangle
will be parallel to the y-axis,
so it might look something like this.
The width of this rectangle is delta x,
which tells us we'll
integrate with respects to x,
as we see here.
So let's go ahead and see
if we can set this up.
We have the volume is equal to two pi.
Since we're integrating
with respects to x,
we know that a would be zero
and b would be two along the x-axis.
Next, we want the radius function.
Well, the radius function
would be the distance
from the axis of rotation to
the center of the rectangle.
It'd be that distance here.
And this distance would just be x.
Times the height of the rectangle,
well the height would be the
distance between the function
and the x-axis, and this
would be equal to y,
and y is equal to x squared plus two.
And then we're integrating
with respects to x.
Let's go and distribute the x.
Looks like we'll have x cubed plus two x.
Now we'll go ahead and
find the anti-derivative.
So we'll have x to the fourth over four
plus two times x squared over two.
I'm going to go ahead and rewrite
this as 1/4 x to the fourth
plus x squared, now we'll
go ahead and evaluate this
at the upper and lower
limits of integration.
So first we'll place x with two.
Then we'll replace x with
zero, both terms would be zero.
So we have two pi times,
this is going to be four plus four,
that would be eight, so this volume
is going to be equal to 16 pi cubic units.
Okay, that'll do it for this first video.
Next, we'll take a look at rotating about
the x-axis using the shell method.
Hope you found this helpful.
