Day 39 in Mat129

Last time: Examples

Next time: Substitution

Announcements

Thu

4/22

Section 5.6

Read 6.1 and 6.2 for Monday

Begin on-line homework "Homework: Substitution" (due 4/30) -- this is our last homework. The rest will be extra credit, which can substitute for other homeworks on which you didn't do as well.

Today: Section 5.6: The substitution rule
- The substitution rule is the chain rule backwards: I'd prefer that we think as follows: ${h(g(b))-h(g(a))=\int_{a}^{b}h'(g(x))g'(x)dx}$ and, even better, ${h(g(b))-h(g(a))=\int_{a}^{b}h'(g(x))g'(x)dx=\int_{g(a)}^{g(b)}h'(u)du}$
  Forgetting for a moment that we might know how to solve this!;), we can always do the change of variables
  ${\int_{a}^{b}f(g(x))g'(x)dx=\int_{g(a)}^{g(b)}f(u)du}$ and hope that the integral on the right is easier to solve (certainly less cluttered).
  Writing it in this last way may be mysterious, because of the change of variable to u (and the change in the limits); but it's the disappearance of g'(x) that's really curious. It falls right out of the change of variables, however:
  ${u=g(x)\Longrightarrow\frac{du}{dx}=g'(x)}$ Solving for du, we get du=g'(x)dx, and that piece disappears.
- Examples:
  - #2, p. 295
  - #7, p. 295
  - #13, p. 296
  - #31, p. 296
  - #41, p. 296
  - #57, p. 296
  - #63, p. 297
  - #64, p. 297

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