Day 24 in Mat329

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Today:

Announcements
- Your homework from 14.8 is returned: Graded #3 and 12
  - Hooray! A student provided a Mathematica solution set! Good work Matthew: beautiful 3D contours!
  - #3 and #12 contrast:
    - #3: looking for a max/min on an unbounded set.
      Good to think in terms of bowls.
    - #12: looking for a max/min on a bounded set. In this case, we have to have one, since $f$ is continuous. A continuous function attains both extremes on a closed and bounded set (the extreme value theorem).
      Watch out for 0 divides! You may lose track of solutions....
- PS: fall break! Have a good one....
Today we continue with integration, section 15.2 (iterated integrals) and section 15.3 (more general areas).
- We ended last class looking at some examples (from 15.2):
  - #1, p. 1011
  - #6
  - #15 (two ways)
  - #26
Section 15.3 is no great shakes. We're just moving beyond rectangles in the plane. 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.
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....
Examples (15.3):
- Let's draw the regions being integrated over for problems #1-6
- #6
- #15
- #17

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