What Happened 11/16

On Thursday 11/16, Dr. Kerckhove lectured in Dr. LeCrone’s absence. The topic of the day was Riemann Integration.

Dr. K began with a definition of Riemann Integration and then went through an outline of what we would cover:

Computing Riemann sums to approximate Riemann Integral
Controlling error
Systematic way to improve a given estimate
If all sequences of improved values converge to the same number, then that number is the Riemann Integration result.
A theorem that characterizes properties of the function f that guarantee steps 1-4 will work to produce exactly one result

We then covered several definitions:

f is bounded on [a,b]
a partition P of [a,b], as well as the set of all possible partitions of [a,b], $\mathcal{P}_{[a,b]}$
A refinement Q of a given partition P
- We looked at the example $f(x)=x\; P=\{0,1/2, 1\},\;Q=\{0,1/3,1/2,5/8,7/8,1\}$ and after examining the right end point sum produced a lemma
- Lemma: Under certain conditions, if Q is a refinement of P, then $|Q_{error}|\leq |P_{error}|$
From a given f,[a,b], P we can construct
1. The Lower Sum for f over [a,b] using P (denoted $L(f,[a,b],P)$ )
2. The Upper Sum (denoted $U(f,[a,b],P)$ )

We then produced the following Lemma:

For a lower sum built from P if we take a refinement Q of P the the lower sum for the refinement satisfies $latex L(f,[a,b],Q)\geq L(f,[a,b],P)$ and $U(f,[a,b],Q)\leq U(f,[a,b],P)$ . Moreover $s=latex \Sigma_{k=1}^n f(x)(x_k-x_{k-1})$ is any Riemann sum built from P $x\in[x_{k-1},x_k]$ , then $L(f,[a,b],P)\leq s \leq U(f,[a,b],P)$ .

A proof of the moreover section of the lemma was then produced.

We made two observations:

For any estimate $|P_{error}|\leq |U(f,[a,b],P)-L(f,[a,b],P)|$
If Q is a refinement of P then $|U(f,[a,b],Q)-L(f,[a,b],Q)|\leq |U(f,[a,b],P)-L(f,[a,b],P)|$