# Type II regions in three dimensions | Divergence theorem | Multivariable Calculus | Khan Academy

Let's now think about Type 2 regions.
And you'll see that they're kind of very similar definitions
and it's really a question of orientation.
Type 2 region is a region-- I'll call it
R2-- that's the set of all x, y's, and z's in three
dimensions such that-- and now instead
of thinking of our domain in terms of xy-coordinates,
we're going to think of them in terms of yz coordinates,
such that our yz pairs are a member of some domain.
I'll call it D2 since we're talking about Type 2 regions.
And x is bounded below by some function of yz.
So I'll call it g1 of yz is less than or equal to x, which
is less than or equal to some other function of yz, g2 of yz.
And so you'll immediately see a very similar way of thinking
functions of x and y as we had in a Type 1 region,
we now have x varying between two functions of y and z.
Now let's think about some of the shapes we explored.
We saw that these two right up here,
this sphere and the cylinder, were Type 1 regions,
but this dumbbell, the way that I oriented it here,
was not a Type 1 region.
Let's think about which of these are Type 2 regions
and what might not be a Type 2 region.
So first let's think about the sphere.
So I have my axes right over here.
Let me scroll down a little bit.
So I got my axes.
And so over here, our domain, we could still
construct our sphere, but our domain
is now going to be in the yz plane.
So yz plane is this business right over here.
So this will be our domain.
I want to make it more spherical than that.
So our domain is this right over here in the yz plane.
That is our D2.
And now the lower bound, in order
to construct the solid region of the sphere or the globe
or whatever you want to call it, the lower bound on x
would be kind of the back half of the sphere, the one that's
away from us right over here.
So the lower bound-- so let me see
how well I can wire frame it at first.
I can do a better job than that.
So with my ghost do something like that,
then do something like that.
But this is if the domain right over here is transparent.
But all we might catch-- we'll just catch a glimpse of it
in the back right over here.
So it's the side of the sphere that's facing away from us.
And then the upper bound on x would
be the side of the sphere that's facing us.
So if I were to do some contours,
it might look something like this
and then look something like this.
And then we would color in this entire region right over here.
And x can take on all of the values above that magenta
surface and below this green surface.
And essentially, it would fill up
the globe for every yz point in our domain.
So a sphere is both a Type 1 and a Type 2 region.
Actually, we're going to see it's
going to be a Type 3 region as well.
Can we construct it or think about it in a way
that it would actually be a Type 2 region?
So let's try to do that.
So let me paste it.
So what if we had a domain-- what
if our domain was something like this?
It was a rectangle in the yz plane.
So this is our domain, a rectangle in the yz plane.
So that would be my D2.
And what if the lower bound was kind
of the back side of the cylinder?
So the backside of the cylinder, try
to draw it as good as I can.
And so if we just saw the outside of it,
it would look something like that.
It's facing away from us so we barely see it.
If we could see through the cylinder
or see through the little flat cut of the cylinder,
it would look something like that.
So that over there would be our g1.
And then our g2 would be the front side of the cylinder.
The g2 could be the front side of the cylinder.
So let me color it in as best as I can.
So the g2 would be the front side of cylinder.
And x can vary above g1 and below g2,
and it would fill up this entire cylinder.
So we see that this same cylinder that we also saw
was a Type 1 region can also be a Type 2 region.
could not be a Type 1 region?
Can this be a Type 2 region?
I'll do it the same way.
We can construct a domain.
So maybe our domain, it's in the y-- well,
it should be in the yz plane if we're talking about Type 2
regions or if we want to think of it as a Type 2 region.
So our domain could be this kind of flat hourglass shape that's
in the yz plane.
So our domain could be a region that looks something
like this in the yz plane.
So this is kind of flattened out.
So this is our domain right over there.
And then the lower bound on x, g1,
could be a surface, the function of y and z
that is kind of the backside of our hourglass.
The backside of our hourglass you could see.
I'll try to show the contours from the underside
right over there.
So that could be our g1.
And then our g2 could be the front side of the hourglass.
So my best attempt to draw the front side of the hourglass.
And I could color it in.
And so the way I somewhat confusingly drew it just now,
you see that this hourglass oriented the way it is
would actually be a Type 2 region.
Now if we were to rotate it like this-- so let
me draw it like this.
Edit.
So if we were to make it like this so
that the top of my hourglass is facing
us-- try my best to draw it.
So let's say the top intersects the x-axis right over there.
This is the bottom of my hourglass right over there.
And then it bends in and then comes back out like that.
For the same reasons that this was not a Type 1 region,
this now would not be a Type 2 region.
For any zy, you can see there could be multiple x points that
are associated with the different points
of this hourglass.
You can't just have a simple lower and upper round
functions right over here.
So this right over here is not a Type 2 region.
You could show a rationale or this
is going to be a Type 1 region.
You could create a region over here in the xy plane
and have an upper and lower bound functions for z.
So you could be Type 1, but this will not be Type 2.