What Are Type I, Type II, and Type III Regions in Calculus 3?

[Medtner]

It's calculus 3 question
My textbook does a horrible job at explaining the differences between Type I and Type II regions for z =ƒ(x,y) functions, and Type 1, Type 2, and Type 3 for B=ƒ(x,y, z). And when integrating over these "D" regions, the method of integration doesn't really seem to be really different at all from normal ∫∫ and ∫∫∫ integrations. The only difference I see is that we're integrating with respect to functions g(x) and h(x) instead of variables. I just need someone to clarify wth my book is trying to show me.

 

[The Bill]

I'm not familiar with that terminology. What textbook is it? If you tell us, I can see if there's a copy in the library I can look at to figure this out.

 

[Mark44]

It's calculus 3 question
My textbook does a horrible job at explaining the differences between Type I and Type II regions for z =ƒ(x,y) functions, and Type 1, Type 2, and Type 3 for B=ƒ(x,y, z). And when integrating over these "D" regions, the method of integration doesn't really seem to be really different at all from normal ∫∫ and ∫∫∫ integrations. The only difference I see is that we're integrating with respect to functions g(x) and h(x) instead of variables. I just need someone to clarify wth my book is trying to show me.

I'm going to guess that your textbook is one by Stewart, who I believe is the only person making these distinctions (Type I, Type II, etc.).
I'll focus on double integrals $\int \int_D f(x, y) dA$. The Type I and Type II business has to do with how the region over which integration is being performed is defined. In a Type I region, $D = \{(x, y) | a \le x \le b, g_1(x) \le y \le g_2(x) \}$. In other words, the inner integral runs from a lowest value of $g_1(x)$ up to $g_2(x)$, and the outer integral runs from x = a to x = b. An example of such a Type I region would be the region bounded by the parobolas $y = x^2 + 1$ and $y = x^2$.

A Type II region is one described by $D = \{(x, y) | c \le y \le d, h_1(y) \le x \le h_2(y) \}$. The region bounded the graphs of $x = y^2$, $x = \frac 1 2 y^2 + 1$, the line y = 1, and the x-axis is a type II region. When you're integrating over this type of region, the inner integral involves a horizontal line running from $h_1(y)$ to $h_2(y)$. The outer integral runs from y = c to y = d.

Some regions can be both types, which means that they can be described in either of the ways I showed above. In questions that ask you to change the order of integration, you are essentially changing from one description to the other. Some integrals can be very difficult or even impossible with one order of integration, but very easy if the order of integration is switched.

Hope that helps...

 

[Medtner]

Although your explanation is similar to the textbook's, you managed to clarify one thing that they pretty much glossed over which helped a lot. Thanks so much, and yes it's Stewart.