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after going through type 1 and type 2 region definitions you could probably guess what a type 3 region is going to be so a type 3 type 3 region is a region in three dimensions I'll just call it our region is our sub 3 since we call 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 YS and Z's such that the X Z pairs X Z pairs are a member of a domain I'll call this domain D 3 D sub 3 and and let's put a comma here 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 less than or equal to Y which is going to be less than or equal to it's gonna Y Y is gonna 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's and so it's going to be in the XZ plane so over here our domain could be this region right over here in the XZ plane so I'll color it in it could be that region right over there in the XZ plane and then the lower bound on Y will be the part that is behind the sphere in this direction right over here is the stuff it actually might be a little bit hard to visualize since I'm redrawing on top of it but and then the upper bound on Y is going to be this side right over here it's going to be this side right over here it's going to be the upper bound so all of this is now is now going to be green let me redraw the sphere just to make it clear so assuming that let me just draw another coordinate axes if you know another coordinate axes the backside of the sphere in the y-direction so I guess let's think of it this way so this is the Hemis 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 wise upper bound that bh2 would be all of this business 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 and let me see how well I can color it and 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 a lower bound on Y so it's gonna 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 we 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 this the cylinder at least the way it was oriented there actually any cylinder actually will also be a type 3 region exact same argument so let me let me draw my axes again 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 XZ plane so our domain can be a rectangular region and the X Z plane just like that and then the lower bound on Y could be d 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 the upper bound on Y would be this side this side right right over there so once again this is also a type 3 region by the same logic this this this one right over here this hourglass can be a type 3 region the front side the front side would be would be this front side right over here all of this including that stuff that I so it'd be all of that and then the back side when you think of in terms of why the backside the backside would be that part right over there right over there and once again once again this could also this could also be a type 3 region the domain the domain let me draw the domain would be kind of this cross 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 hourglass and the upper bound on Y would be the front half would be the front half right let me do that in magenta color because I've been using that or actually that green color so the upper bound on Y would be this so I guess you could say this right half right over here so what would not be a type 3 region well if we just rotated this around like that so let me just draw something that is not a type 4 the region just to show you that this definition does not include everything so something that would not be a type 4 reason for the same argument as we've seen before for type 2 and type 1 regions 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 hourglass an hourglass that looks like this 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 break up this region into two type three regions but the whole thing itself is not a type 3 region so not not type 3