Day 24 in MAT229

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Announcements:
- Here's a recording of Tuesday's Lab
  Thursday's Zoom recording.
- 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[]
```
- A reminder that you can Zoom into these labs. You're welcome to come virtually anytime.
  For all of you here today, you're welcome to zoom in as well, so that we can share screens if some problem arises, or if we want to show off something you've done.
- We're in week seven, and 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 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 we introduce "improper integrals" -- our first serious foray into the infinite.
Today: lab 7 (review, for the most part; but there's a twist to integrating improper integrals numerically)
- This week we're looking into improper integrals: that is, integrals that have an infinity causing us trouble: either staring us in the face, e.g. \[ \int_{1}^{\infty}\frac{1}{x^2}\ dx \] or cleverly concealed inside the integrand (which blows up to $\infty$ at an endpoint, which is the case here, or at some point on the interior of the interval of integration): \[ \int_{0}^{1}\frac{1}{\sqrt{x}}\ dx \]
- It turns out that we can't (yet) do anything about that integral running over an infinite interval, but we can handle that second problem numerically: of an integrand with a vertical asymptote at an endpoint.
  But we'll see that only certain of our numerical integration schemes are helpful in this case. Just one of the exciting problems awaiting you in Lab 7!
- Lab 7 (.nb) (a pdf version)
- 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.

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