INSTRUCTOR: Consider the region bounded by the curves, y equals natural log of x, and y equals e to the x in the interval, closing interval from 1 to 2. Decompose the region into smaller regions of Type II. First, I would encourage you to look at the gravs of these two functions on that interval, as that will help you determine what those smaller regions are. Now, because we're talking about Type II regions, we first consider boundaries on the y variable, and then consider the corresponding boundaries in our region in terms of x.

So our regions will be, for the first smaller region, is the set of ordered pairs, x comma y, such that 0 is less than or equal to y, is less than or equal to 1. And 1 is less than or equal to x, which is less than or equal to e to the y. Now, there are actually going to be three such regions. So I'm going to union each of these sets together.

Set 1 unioned with the set of all ordered pairs, x,y such that 1 is less than or equal to y, is less than or equal to e. And 1 is less than or equal to x, is less than or equal to 2, unioned with our third region, set of ordered pairs xy such that e is less than or equal to y, it's less than or equal to e squared. And natural log of y is less than or equal to x, is less than or equal to 2.