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# Shell method with two functions of y

Video transcript

I'm going to take the region
in between the two curves here, between the yellow curve--
defined as a function of y as x is equal to y minus 1
squared-- and this bluish-green looking line-- where y
is equal to x minus 1. So I'm going to take this
region right over here, and I'm going to rotate
it around the line, y equals negative 2,
to get this shape that looks like the front of a jet
engine or something like that. And we're going to want to
figure out what its volume is. And you can actually
approach this with either the disk
method or the shell method. In the disk method,
you would create disks that look like this. And you would be doing
integrating with respect to x. You will have to break up
the problem appropriately, because you have a
different lower boundary. You would have to break
this up into two functions, an upper function
and a lower boundary for this interval in x. And then a different one
for that interval in x. But you could use
the disk method. But instead of
that, we don't feel like breaking up the functions
and doing all of that. So we're going to do
it in the shell method, especially because we've already
expressed one of our functions as a function of y. And this one won't
be too hard to do. So what we're going
to do, once again, is imagine constructing
these little rectangles that have height dy. And what we're going to do
is rotate those rectangles around the line y
equals negative 2. So let me draw that same
rectangle over here. And when you do that,
you construct a shell. So let me do that. So, go between these two points. And then this is what it would
look like when it's down here. And then, let me make it
clear that this constructs a shell of thickness
dy or of depth dy. So let me do it like that. So that's my shell, and it
has some thickness to it. And that thickness is dy. So that's the
thickness of my shell. And let me shade
it in a little bit. Make sure you can see the
3-dimensionality of my shell. So like we've done with
all of these problems, our goal is to really
just figure out what the volume
of each shell is. And then we can enter it for
a specific y in our interval. And then we integrate along
all the y's of our interval. And we've done this
many times before. The first thing
we think about is what's the radius
one of these shells? So what I'm doing
right here in magenta, what is the radius of
something like that? Well, it's essentially
going to be the distance between y
is equal to negative 2 and our y value for
that specific y. So this distance
right over here is y. And then we're going
to have another 2. So the whole distance
is going to be y plus 2. Another way of thinking
about it is this is essentially y
minus negative 2 to get the distance, which
is going to be y plus 2. So the radius of one
of our little shells is going to be y plus 2. If the radius is
y plus 2, then we know that the circumference
of this circle right over here is going to be 2
pi times y plus 2. And then the surface area,
the outside surface area, of the shell, the
stuff out here, is just going to be that
circumference times I guess you could say the width
of this shell times this distance right over here. Or we could say times this
distance right over here. And what is that distance? Remember, we want everything
expressed as a function of y. Well, it's going to be the upper
function as a function of y minus the lower function. And we think about
the upper function, it's the function
that's giving us higher x values
in that interval. And so this blue function
is the upper function when we think in terms of y. But we have to express
it as a function of y. So let me do that. So we can rewrite this
as add 1 to both sides, you get x is equal to y plus 1. So that is our upper function. And then this is
our lower function. If you were to tilt
your head to the right and look at it that
way, you'll see that this will be
the upper function, and this will be
the lower function for the same value of y. This gives a higher value
of x than this one does for the same value of y. It gives the upper x values. So, the area is going
to be the circumference times this dimension. Let me write this. The area of one of those shells
is going to be 2 pi times y plus 2 times the distance
between the upper function. So the distance between the
upper function y plus 1, x is equal to y plus 1,
and the lower function, x is equal to y minus 1 squared. I'll put the parentheses
in that same color. And then if we want the
volume of that shell, so we've got the outside surface
area of the shell right now. We just multiply it by its
depth, which is just dy. So that sets up our integral. The volume of one shell--
I'll do it all in one color now-- is 2 pi times y plus 2
times y plus 1 minus y minus 1 squared. Then we multiply that times
the depth of each shell, dy. And then we integrate
over the interval. So the volume is
going to be this. And what's the interval? You might be able to eyeball it. But we can actually
solve that explicitly. When do these two
things equal each other? Well, you could
just set y plus 1 to be equal to y minus 1
squared, so let's do that. So if we set y plus 1,
this, to be equal to that, it's going to be equal
to y minus 1 squared. So let me expand that out. That's going to be y
squared minus 2y plus 1. And let's see. I could subtract
y from both sides and I could subtract
1 from both sides. Minus y, minus 1. And then I am left with,
on the left hand side, 0. And on the right hand side,
I have y squared minus 3 y. And that's it, plus 0. So 0 is equal to
y times y minus 3. So the zeros of
this, when these are equal are when y is equal
to 0 or y is equal to 3. And we see that right over here. When y is equal to 0, these
two functions intersect. And when y is equal to 3,
these two functions intersect. So our interval is going
to be from y is equal to 0 to y is equal to 3. So using the shell
method, we have been able to set up
our definite integral. And now we can think about how
we can evaluate this thing.