# How to Remember Type I, II, and III Regions in Multivariable Calculus

*Type I pairs with the variable that runs vertically in the usual representation of the coordinate system. The remaining types are paired with the rest of the variables in ascending order.*

I’ve always had trouble remembering the definitions of type I, II, and III regions. In particular, it’s tricky to remember which type (I, II, III) is paired with which axis ($x,$ $y,$ $z$).

In $2$-dimensional space, type I and II regions are paired with the $x$ and $y$ axes, respectively:

- A region is of
*type I*if it can be decomposed into vertical segments parallel to the*$y$-axis*, with each $x$-value having at most $1$ segment and the segment endpoints tracing out continuous functions. - A region is of
*type II*if it can be decomposed into horizontal segments parallel to the*$x$-axis*, with each $y$-value having at most $1$ segment and the segment endpoints tracing out continuous functions.

But in $3$-dimensional space, type I, II, and III regions are paired with the $z,$ $x,$ and $y$ axes, respectively:

- A region is of
*type I*if it can be decomposed into vertical segments parallel to the*$z$-axis*, with each $xy$-value having at most $1$ segment and the segment endpoints tracing out continuous functions. - A region is of
*type II*if it can be decomposed into horizontal segments parallel to the*$x$-axis*, with each $yz$-value having at most $1$ segment and the segment endpoints tracing out continuous functions. - A region is of
*type III*if it can be decomposed into horizontal segments parallel to the*$y$-axis*, with each $xz$-value having at most $1$ segment and the segment endpoints tracing out continuous functions.

It seems as though these definitions aren’t even consistent with each other! Type I pairs with $y$ in $2$-dimensional space, but with $z$ in $3$-dimensional space.

Further complicating the matter, they aren’t even consistent in being inconsistent! Type II pairs with $x$ in $2$-dimensional space, and again with $x$ in $3$-dimensional space.

However, I recently noticed a trend that seems to generalize well:

- Type I pairs with the variable that runs vertically in the usual representation of the coordinate system.
- In $2$-dimensional space, that's $y,$ and in $3$-dimensional space, that's $z.$

- The remaining types are paired with the rest of the variables in ascending order.
- In $2$-dimensional space, we've already assigned $y$ to type I, and the only remaining variable is $x,$ so we assign it to type II.
- In $3$-dimensional space, we've already assigned $z$ to type I, and the remaining variables in alphbetical order are $x,y,$ so we assign $x$ to type II and $y$ to type III.

Generalizing to $N$-dimensional space with coordinates $(x_1, x_2, x_3, \ldots, x_N),$ we have the following pairings:

- Type I is paired with $x_N$
- Type II is paired with $x_1$
- Type III is paired with $x_2$
- Type IV is paired with $x_3$
- (and so on)