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In this and the
next few videos, I hope to explore different types
of regions in three dimensions. And these will be
useful for thinking about how to evaluate different
double and triple integrals and also some interesting proofs
in multivariable calculus. So the first type of region,
and it's appropriately named, we will call a type 1 region. At first, I'll give
a formal definition. And hopefully, the
formal definition makes some intuitive sense. But then I'll draw a
couple of type 1 regions, and then I'll show
you what would not be a type 1 region
because sometimes that's the more important question. So type 1 region, maybe a type
one region R, is the set-- and these little curly
brackets means set-- is the set of all
x, y's, and z's. It's the set of all
points in three dimensions such that the x and y's
are part of some domain, are a member-- that's what
this little symbol represents-- are a member of some domain. And z can-- essentially
varies between two functions of x and y. So let me write it over here. So f1 of x, y is kind
of the lower bound on z. So this is going to be less
than or equal to z, which is less than or equal to
another function of x and y, which is going to be less than
or equal to f2 of x and y. And let me close
the curly brackets to show that this was all a set. This is a set of
x, y's, and z's. And right here, we
are defining that set. So what would be a
reasonable type 1 region? Well, a very simple type
one region is a sphere. So let me draw a
sphere right over here. So in a sphere, where it
intersects the x, y plane-- that's essentially this
domain D right over here. So I'll do it in blue. So let me draw my best
attempt at drawing that domain so that this is the domain D
right over here for a sphere. This is a sphere
centered at 0, but you could make the same argument
for a sphere anywhere else. So that is my domain. And then f1 of x, y, which
is a lower bound of z, will be the bottom
half of the sphere. So you really can't see
it well right over here, but it would be-- these
contours right over here would be on the bottom half. And I can even color in
this part right over here. The bottom surface of our
sphere would be f1 of x, y, and f2 of x, y as
you could imagine, will be the top half of the
sphere, the top hemisphere. So it'll look
something like that. This thing that I'm
drawing right over here is definitely
a type 1 region. As we'll see, this could
be a type 1 type 2, or a type 3 region. But it's definitely
a type one region. Another example of a type 1
region-- and actually this might even be more obvious. So let me draw some
axes again, and let me draw some type of a cylinder. Just to make it clear that
our domain, where the x, y plane does not have to
be inside of our region-- let's imagine a cylinder
that is below-- well, actually I'll draw it above--
that is above the x, y plane. So this is the bottom
of the cylinder. It's right over here. And once again, it
doesn't have to be centered around the z-axis. But I'll do it that way
just for this video. Actually, I could draw it a
little bit better than that. So this is the bottom
surface of our cylinder, and then the top
surface of our cylinder might be right over here. And these things actually
don't even have to be flat. They could actually
be curvy in some way. And in this situation,
so in this cylinder-- let me draw it a
little bit neater. In this cylinder
right over here, our domain are all of the values
that the x and y's can take on. So our domain is going
to be this region right over here
in the x, y plane. And then for each of that,
those x, y pairs, f1 of x, y defines the bottom
boundary of our region. So f1 of x, y is going to
be this right over here. So you give me any of these
x, y's in this domain D, and then you evaluate the
function at those points, and it will correspond to
this surface right over here. And then f2 of x, y,
once again, give me any one of those x, y
points in our domain, and you evaluate
f2 at those points, and it will give you
this surface up here. And we're saying
that z will take on all the values in between,
and so it is really this whole solid-- it's really
this entire solid area. Likewise, over
here, z could take on any value
between this magenta surface and this green surface. So it would essentially
fill up our entire volume so it would become
a solid region. Now, you might be wondering what
would not be a type 1 region? So let's think about that. So it would essentially
be something that we could not
define in this way, and I'll try my best to draw it. But you could
imagine a shape that does something funky like this. So there's like
one big-- I guess you could imagine a
sideways dumbbell. So a sideways
dumbbell-- and I'll maybe curve it out a little bit. So this is the kind of the top
of the dumbbell-- or an hour glass, I guess you could
say, or a dumbbell. It would look
something like that. So I'm trying my
best to draw it. It would look
something like that. And the reason why this is
not definable in this way-- it becomes obvious if you kind of
look at a cross section of it. There's no way to define
only two functions that's a lower bound and an
upper bound in terms of z. So even if you say, hey, maybe
my domain will be all of the x, y values that can
be taken on-- let me see how well I can draw this. So you say my x,
y values-- so let me try to draw this whole
thing a little bit better, a better attempt. So you might say, OK, for
something like a dumbbell-- let me clear out
that part as well. For something like a dumbbell--
so let me erase that. So for something like
a dumbbell, maybe my domain is right over here. So these are all the x, y
values that you can take on. But in order to have a
dumbbell shape, for any one x, y, z is going to
take on-- there's not just an upper and a lower
bound, and z doesn't take on all values in between. Well, let me just draw
it a more clearly. So our dumbbell-- maybe
it's centered on the z-axis. This is the middle
of our dumbbell, and then it comes out like that. And then up here, the z-axis--
so it looks like that. And then it goes
below the x, y plane, and it does kind
of a similar thing. It goes below the x, y plane
and looks something like that. So notice, for any
given x, y, what would be-- if you attempted
to make it a type 1 region, you would say, well, maybe
this is the top surface. And maybe you would say down
here is the bottom surface. But notice, z can't take
on every value in between. You kind of have
to break this up if you wanted to be able
to do something like that. You would have to break this
up into two separate regions where this would be
the bottom region, and then this right over here
would be another top region. So this dumbbell shape itself
is not a type 1 region, but you could
actually break it up into two, separate
type 1 regions. So, hopefully, that helps out. And actually, another
way to think about, this might be an
easier way-- if we were to look at it
from this direction, and if we were to just
think about the z, y, if we were just
thinking about what's happening on the z, y
plane-- so that's a z, and this is y right over
here-- our dumbbell shape would look something like
this, my best attempt to draw our dumbbell shape. And so if you get a given
x or y, maybe x is even 0, and you're sitting right here
on the y-axis, notice z is not, even up here, cannot be
a function of just y. On this top part, there's
two possible z-values that we need to take
on for that given y-- two possible z-values
for that given y. So you can't define
it simply in terms of just one lower bound function
and one upper bound function.