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The author makes a great deal of "Type I" (p. 1013) and "Type II" (p. 1014) regions. This is a mere convenience: each has a pair of "rectangular" edges, in either $x$ or $y$ directions. Turn your book sideways.
It's no big deal to turn a general region into such regions -- the question is whether you need to, and how, if so.
The essential point is this: we're really trying to do is parameterize the boundaries in a sensible way, and, if we can, then we can go back to Fubini's theorem and do twice as much work as we did when we did univariate integrals.
Take a look at Figure 18, p. 1018. We can break it down into a pair of these types of regions, and move ahead....
There are a few properties of integrals worth paying attention to in this section, such as the inequality at the bottom of 1018. These should make intuitive sense to you -- hopefully so!