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Current time:0:00Total duration:8:39

Video transcript

and this in 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 type type 1 region it first I'll give a formal definition 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 may be a type 1 region R is the set and these little curly brackets means set it's the set of all X YS and Z's it's a set of all points in three dimensions such that the X and YS are part of some domain so the X and Y's are part of some domain are a member that's what this little symbol represents our member of some domain and Z can essentially varies between two functions of x and y so let me write it over here so f 1 of X Y is kind of the lower bound on Z so this is going to be less than or equal to Z less than or equal to Z which is less than or equal to the another function of x and y which is going to be less than or equal to F 2 of X and y and let me close the curly brackets to show that this was all upset this is a set of XY z-- and Z's and right here we are defining that set so what would be a reasonable type 1 region well the very simple type 1 region is a sphere so let me draw a sphere right over here so in a sphere where it intersects the xy-plane that's essentially this domain D right over here so I'll do it in blue so let me draw 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 can make the same argument for a sphere anywhere else so that is my domain and then F 1 of XY which is a lower bound Z will be the bottom half of the fear so you really can't see it well right over here but it would be 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 is you can imagine will be the top half of the sphere the top hemisphere so it'll look it'll look something something like something like that so that is that right over this thing that I'm drawing right over here is definitely a type 1 region as well see this could be a type 1 type 2 or type 3 region but it's definitely it's definitely a type 1 definitely a type 1 region another example of a type 1 region and actually this might even be more obvious so let me put some draw some axes again and let me draw let me draw some type of a cylinder and I'm not going to make just to make it clear that our domain where the xy-plane does not have to be inside of our region let's imagine a cylinder that is below that is well actually I'll draw it above that is above the XY 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 I can draw it a little bit better than that so this is the bottom 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 don't have to be 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 and this cylinder right over here our domain 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 XY plane and then for each of that those XY pairs F 1 of XY defines the bottom boundary of our region so f 1 of XY is going to be this right over here so you give me any of these X YS in this domain D and it course and then you evaluate the function at those points and it will correspond to this surface right over here and then F 2 of X Y once again give me any one of those XY points in our domain and it you evaluate f2 at those points and it will give you this surface up here and we're saying that Z is and Z will take on all the values in between and so it is really this whole solid it's really this entire 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 you can imagine a shape that does something funky like this so there's like one big I guess you could imagine a a sideways dumbbell so a sideways dumbbell I'll maybe a curve it out a little bit so maybe it's so this this is kind of the top of the dumbbell and then it or an hourglass I guess you could say or a dumbbell it would look something like that so I'm trying my best to draw it it looks something like that and the reason why this is not 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 there's no way to define only two functions that's a lower bound and an upper bound in terms of Z so let me so even if you say hey maybe my domain 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 XY values let me try to draw this whole thing a little bit better better attempt so you might say okay for something like a dumbbell clear out to this that part as well for something like a dumbbell so let me erase that so for something like a dumbbell maybe my domain is 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 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 in between let me just draw it 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 XY plane it does kind of a similar thing it goes below the XY 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 this is the top surface and maybe you would say down here is the bottom surface 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 would be 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 it this might be an easier way if we just draw if we were to look at it from this direction and if we were to just think about the zy if we were just think about what's happening on the zy plane so that's Z and this is Y right over here our dumbbell shape our dumbbell shape would look something like this 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 4 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