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### Course: Multivariable calculus>Unit 5

Lesson 10: Types of regions in three dimensions

# Type I regions in three dimensions

Definition of Type 1 regions. Visualizing what is and isn't a Type I region. Created by Sal Khan.

## Want to join the conversation?

• So, basically, a Type I region passes the vertical line test for each of it's bounding functions...right??
• Yes; functions by definition pass the vertical line test. In the case of 3-variable functions: for any combination of x and y, there is a unique z value assigned to it. In other words, a function is defined so that there is only one height (z) for every point in 2-D space (x,y). Therefore a vertical line will only pass through the surface once. Type I regions are bounded by functions of (x,y), so each boundary passes the vertical line test.

N.B. It follows that any vertical line passing through a Type I region will cross its boundary surface exactly twice (once in & once out).
• I don't get is why the dubmbell is not a type I region.
• Great question. I'm also unsure of why that is the case, but here is hopefully a good enough explanation.

We begin with the definition of a type I region; `(x, y, z)` means that it is a collection of points in 3D space, or simply a 3D figure with volume. `(x, y) ϵ D` simply means that x and y are part of a domain, or that the region is not infinite.

Now comes the hard part to understand. `f1(x, y) ≤ z ≤ f2(x, y)` put into words is simply that for all z, there is a surface (because f(x, y) is a surface) f1 that is the lower bound and f2 that is the upper bound.

Using the examples, we see that for the sphere, the upper bound is the green hemisphere, whereas the lower bound is the purple hemisphere, which are both surfaces that bound all z. Likewise, for the cylinder, the green circle and purple circle are the bounds.

For the dumbbell, there cannot be a lower or upper bound that can completely bound or "wrap around" the 3D figure.

Another thing to note is that because the bounds are surfaces, for any (x, y) (which can be imagined as a vertical line parallel to the z-axis), there can be at most 2 values of z where it crosses the boundary surfaces. This can be considered the 3D version of the vertical line test of sorts.

I'm unsure if this is correct, but hopefully I helped.
• Is a surface like the plane z = 3x + 5y + 2 a Type I reigon
• No, because a plane is a single surface. A Type I region is a region between two surfaces.
• Ok so he is saying that dumbbell wouldn't fit into type 1 but how come sphere applies to that? Y values also changes when you go from top the bottom(for sphere)
• For the sphere, if you fix an (x, y) inside its D, there are two points on the sphere's boundary, with those (x, y) coordinates: the one above it, and the one below it. For the dumbbell, that is not the case. There are places (x, y) inside its D where anywhere from two to four points share the same (x, y) coordinates. So it fails to be a Type I region. Hope that helps!
• Could you please explain how exactly to break up the dumbbell region to make it type 1 region? At Sal says that the UPPER part becomes BOTTOM region, and vice versa... I don't get that and how it makes the dumbbell shape into two separate type 1 regions?
• Good Call.
I don't think Sal was as clear as he could have been on that.
Since the shape is symmetrical above and below the y axis, he was trying to show that the bottom part could be flipped over so that the narrow part, which is the top of the bottom part, is the same as the bottom of the top part. In essence then, you only need to integrate the top part and multiply the result by 2, since the volume of the top half and bottom half are the same.
(1 vote)
• For the dumbbell example, you could just knock the dumbbell on its side (rotate it 90 degrees about either the x or the y axis). Then you would have two surfaces meeting the criteria. but that is just a matter of what you choose to label the coordinates. Is the type of the region dependent on the choice of coordinate orientation?
• I believe it does matter on your coordinate choice (otherwise everything would be rather arbitrary).
(1 vote)
• I'm using his coloring scheme to understand the lower bounds and upper bounds - at , is there is an accidental switch in the colors (purple was the lower bound and green was the upper bound before)?
• Yes, he makes careless mistakes like that all the time.
• Could this be considered to be topology?