Day 26 in MAT229

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Announcements:
- Some comments on Lab 6:
  1. You generally made the right choice for $u$ and $dv$ (or, as I prefer, $f$ and $g$): $f(x)=(\ln(x))^2$ and $g'(x)=x$.
    With this choice, you're one more integration by parts from the answer.
  2. The best choice for $u=4-x^2$. You can substitute for the extra $x^2=4-u$.
    The trig substitution is a lot more work! But it works....
    No one commented on the fact that the left and right rectangle rules and trapezoidal rule all gave the same answer! Is this mysterious? NO!
    The function $f(x)$ was equal at the two endpoints: $f(a)=f(b)$. In this case, the LRR and RRR give the same values, so their average is the same! No mystery....
  3. When you do "Simpson's averaging", you need to remember that \[ S_{2n}=\frac{2M_n+T_n}{3} \] In other words, you get twice the $n$ for the buck. The reason is this: you're using $n+1$ points from the trapezoidal rule, and the $n$ points from the midpoint rule, for a total of $2n+1$ points (and $2n$ subintervals). So \[ S_{200}=\frac{2M_{100}+T_{100}}{3} \]
- Caesar Bao recommends the following commands to auto-save your Notebook -- to help you recover more easily from a crash:
```
SetOptions[$FrontEndSession, NotebookAutoSave -> True]
NotebookSave[]
```
- It's about time for another exam.
  - The exam will be available from 8pm on Friday to 8pm on Sunday.
  - It will be like the last: a combo of IMath, and something to do on Canvas.
  - Again, we're thinking of this as an hour exam -- although we may give you a little extra time, in case you need it. But that's how we're building it.
  - For the Canvas portion you'll be allowed a one-page formula sheet, and your own calculator.
  - The Math & Stat Lab is available for tutoring (virtual). Something to consider. It's free, and features a bunch of our best math/stat majors.
- This week's topic of "improper integrals" is not included on the exam.
Today: Review
- Well, I received absolutely no suggestions of topics or ideas or concepts or problems to go over -- so I guess I won't! I will be available at our office hour, and I'm also available various other times during the day.
- I do want to mention, however, something that I told the Thursday crowd -- that the Tuesday crowd might not have heard.
  All these crazy integration schemes boil down to the following commands in Mathematica: Let's do n=10 intervals (for all but Simpson's, which will be 20!)
```
f[x_]:=Cos[Sqrt[x^3+1]]
a=0
b=4
n=10
dx=(b-a)/n

lrr=dx*Sum[f[a+(k-1.0)dx],{k,1,n}]
rrr=dx*Sum[f[a+(k-0.0)dx],{k,1,n}]
mid=dx*Sum[f[a+(k-0.5)dx],{k,1,n}]
trap=(lrr+rrr)/2
simp=(2*mid+trap)/3
```
  Simpson's uses twice the "$n$", because it uses all $n$ of midpoints $x$-values, and all $n+1$ of trapezoidal's values; so it uses $2n+1$ points, for $2n$ subintervals.
Links:
- Nothing new to report.

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