Day 06, Mat227

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Announcements:
- Your first assignment is returned: randomly selected were
  - 5 - Illustration of the three fundamental rectangle methods.
    - Make sure that you represent each situation graphically.
    - Mid is not the average of LRR and RRR -- trap is!
    - Some of you, in d), just wrote "midpoint" (without justification). Justify yourselves!
  - 16 - focus on using different widths. You have some freedom to choose your method. The best answers used right, left, and ended with trap. Straight to trap would also be very good (because the velocity is presumably steadily increasing).
  - 22 - Cody made an excellent point after class last time -- there are multiple answers!
  - Remember: show your work.
- You have an assignment due Wednesday (for 4.2). We'll do some exercises from section 4.2 today, then I'll let you ask any questions about 4.2's homework.
Section 4.2: The definite integral
- This section generalizes our intuition and examples from the preceding section.
  - There's a lot of notation. Try not to let that intimidate you. A lot of Greek letters. A lot of indices, like i. There are parameters floating around, like n.
- Last time we got to a point at which we had a thing, called a "Riemann sum", based on the following stuff:
  - a function ${f}$ ;
  - an interval ${[a{,}b]}$ over which to calculate the area;
  - a "partition" P of the interval into N subintervals, with "endpoints" ${x_i}$ and widths ${\Delta{x_i}}$ ;
    i is a variable (an index) that runs from 1 to N. It indicates the particular subinterval (the i^th subinterval) under consideration.
    So the endpoints of the i^th subinterval are ${x_{i-1}}$ on the left, and ${x_i}$ on the right.
  - A set C of "Centers" ${c_i}$ of the subintervals, at which we calculate the function values ${f(c_i)}$
    "Centers" might be a poor choice of words, since the centers may not be at all in the centers of the subintervals.
  From this stuff we construct the appropriate sum:
  
  where P is a partition of the interval ${[a{,}b]}$ :
  ${a=x_0<x_1<x_2< \ldots < x_N=b}$
  and where C is a set of centers,
  
  ${C=\{c_1,{\ldots},c_N}\}$
  such that each center falls in one subinterval, where i runs from 1 to N:
  
  ${c_i\in[x_{i-1},x_i]}$
  Last time we saw that a rectangle method dictates a choice of "centers" for the Riemann sum:
  - LRR: ${c_i=x_{i-1}}$
  - RRR: ${c_i{=}{x_i}}$
  - MID: ${c_i{=}{\frac{{x_i+x_{i-1}}}{2}}}$
    (this one really is in the center of the subinterval -- that's the average of the two endpoints).
- If the size of the intervals goes to zero, then the approximations get better and better, until they're perfect!
  
  where
  ${||{P}||}$ is the width of the largest subinterval, ${max(\{\Delta{x_i}|i\in\{1,\ldots,N\}\})}$ for the partition.
  
  The maximum subinterval width ( ${||{P}||}$ ) goes to zero. The partition gets thinner and thinner, as the number of subintervals (n) goes to ${\infty}$ .
- In particular, if ${f}$ is continous on ${[a{,}b]}$ , then ${f}$ is integrable.
  If it's continuous except for a finite number of jump discontinuities, it's still integrable.
  So lots of important functions are integrable.
- Some Important Properties of the Definite Integral (there are more in your textbook!):
- Here's how we'd rewrite our generalization that Area under the graph of f from a to b =
  
  ${{\bar{f}}(b-a)}$ ${\int_{a}^{b}f(x)dx=\bar{f}(b-a)}$ where by ${\bar{f}}$ we mean the average value of ${f}$ .
  Again, we can use this relationship to define the average value of a function on the interval ${[a{,}b]}$
- Examples:
  - Linear and quadratic functions
  - #10, p. 307 (standard stuff, such as we did in 4.1 -- only now we're thinking of it as the definite integral of a function).
  - #33 -- integrals can sometimes be done by inspection!
  - #21 -- this one involves some ugly sums....
  - Let's just glance at the comparison properties of the Integral (box, p. 305), so make sure that they "mak snese".
Section 4.3: the Fundamental theorem of calculus
- Well folks, here it is: the fundamental theorem, that ties together integral and differential calculus. It comes in two parts:
  1. If ${f}$ is continuous on ${[a{,}b]}$ , then
    ${\int_{a}^{b}f(x)dx{=}F(b)-F(a)}$ where ${F}$ is any antiderivative of ${f}$ ; that is, a function such that ${F'=f}$ .
  2. If ${f}$ is continuous on ${[a{,}b]}$ , then the function ${g}$ defined by
    
    ${g(x)=\int_{a}^{x}f(t)dt \;\;\;\;\; a \le x \le b}$
    is continuous on ${[a{,}b]}$ and differentiable on ${(a{,}b)}$ , and ${g'(x){=}f(x)}$ .
  The Fundamental Theorem of Calculus says that in order to integrate, all you have to do is think of an antiderivative F of the integrand f.
- Let's derive/motivate the FTC using the linearization.
- Examples:
  - #3, p. 318 (don't forget about thinking of integrals as areas).
  - #8
  - #13 (a composition, requiring the chain rule)
  - #22
  - #23
  - #58

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Updated on 01/26/2015 18:49:22