Type III regions in three dimensions

After going through type 1 and type 2 region definitions, you can probably guess what a type 3 region is going to be. So a type 3 is a region in three dimensions. Since we called the other the type 2 region R sub 2 and the type 1 region R sub 1, I'll call this region R sub 3-- R with a subscript 3. It's going to be the set of all points in three dimensions. The set of all x, y and z's such that the x, z pairs are member of a domain, I'll call this domain D sub 3. And y is going to vary between two surfaces that are functions of x and z. So y is going to be greater than or equal to the surface-- I'll call it h1 of x, z is going to be less than or equal to y. y is going to be bounded from above by the surface h2 of x, z. And once again, let's close our set notation. So let's think about whether some of these regions that we already saw were type 1 and type 2, whether they're type 3. And then think about what would not be a type 3 region. So let's go to this sphere. What could be our domain? Well the domain is a set of x, z. It's just going to be the in x, z plane. So over here our domain could be this region right over here in the x, z plane. So I'll color it in. It could be that region right over there in the x, z plane. And then the lower bound on y will be the part that is behind the sphere in this direction right over here. It's the stuff-- actually, this might be a little bit hard to visualize since I'm redrawing on top of it. And then the upper bound on y is going to be this side right over here. So all of this is now going to be green. Let me redraw this sphere just to make it clear. Let me just draw another coordinate axes. So another coordinate axes. The backside of this sphere in the y direction. So I guess let's think of it this way. This is the hemisphere mark, this is the halfway point for my sphere. And once again, that's kind of the boundary of our domain. And then the backside-- let me do the front side first. The front side y is upper bound, that would be h2, would be all of this business right over here. So this would be h2 that I'm coloring in. h2 would be the side that is facing in that direction. Let me see how well I can color it. That didn't do a good thing. So h2 is all of this stuff out on this side of the sphere. And then h1 was the lower bound on y. So it's going to be that side right over there. And I could draw it probably a little bit neater. But hopefully you get the point, it would be all of that side. And then y can vary between those two and essentially fill up the region. Make the exact same argument with the cylinder. The cylinder can be defined the same way. So first of all, the sphere is a type 1, type 2, and type 3 region. It meets all of the constraints. The cylinder, at least the way it was oriented there, actually any cylinder, will also be a type 3 region. Exact same argument. So let me draw my axes again. And here to make the cylinder that we've been making, our domain could be a rectangle in the x, z plane. So our domain can be a rectangular region in the x, z plane, just like that. And then the lower bound on y could be that side of the cylinder. The side facing in that direction right over there, that is the lower bound. And then the upper bound on y could be the side facing in that direction. So once again, this is also a type 3 region. By the same logic, this one right over here, this hourglass, can be a type 3 region. The front side would be this front side right over here, all of this, including that stuff that I-- And then the backside when you think of it in terms of y, the backside would be that part right over there. And once again, this could also be a type 3 region. The boundary of the domain would be this cross section right over here. So the boundary of our domain could be that cross section right over there. The lower bound on y would be the back half, the back half of this hour glass. And the upper bound on y would be the front half. Let me do that magenta color because I've been using that. Or actually that green color. So the upper bound on y would be this right half right over here. So what would not be a type three region? Well, if we just rotated this around like that. So let me just draw something that is not a type 3, just to show you that this definition does not include everything. So something that would not be a type 3 region-- for the same argument as we've seen before for a type 2 and type 1 region is an hourglass where it's along the y-axis or at least it's oriented this way. It actually does not have to be right centered on the y-axis. But an hour glass that looks like this, now all of a sudden the y can't be just expressed as being between two surfaces that are functions of x and z. You would have to break this up in order to do that. But you could break up this region into two types 3 regions. But the whole thing itself is not a type 3 region. So not type 3.