Day 4, Mat227

• Reminder: turn off your rectangles.
• Your first assignment is due Friday.

• I ran into a buzz saw when trying to get someone hired for the calc lab after 3:00. There are several issues, but what I'd need to get more hours assigned is a demonstrable commitment to hang around.

  There are other options (for those who can't make calc lab hours): for example, The Math Center has hours outside of the calculus lab.

  You need to make an appointment to use that service, however.

We'll start by finishing off problem #29, p. 295:

Are there some problems that you'd like to look at from section 4.1?

• This section generalizes our intuition and examples from the preceding section:
  • First of all, we can't expect our function to be positive all the time, so we define "signed area" (to include "negative area"):
    

    and then we give a funny name to this area:

    
    

  • Just as before, the area can be calculated by chopping the interval [a,b] up into smaller and smaller chunks, and then using rectangular approximations to the small areas.
    • In our work so far, the intervals into which the interval is chopped have been equal-sized: no reason to stick with that.

      In fact, we talked about why you might want to use rectangles of different widths: when the derivative is large in size, we might want to use smaller (thinner) rectangles, and hence use more of them in a given amount of space.

    • So the essential things we need to construct what's called a "Riemann sum" (rectangular) approximation to the signed area is this:
      • a function ;
      • an interval [a,b] over which to calculate the area;
      • a "partition" of the interval into N subintervals, with "endpoints" and widths ;
      • "Centers" of the subintervals, at which we calculate the function values
      Then we construct the appropriate sum:
      

      where P is a partition of the interval [a,b]:

      

      and where C is a set of centers,

      

      such that each center falls in one subinterval, where i runs from 1 to N:

      

      Question: what are the choice of "centers" for the

      • LRR?
      • RRR?
      • MID?
      In other words, how would you write the in terms of the ?

    • If the size of the intervals goes to zero, then the approximations get better and better, until they're perfect!
      

      where

      is the width of the largest subinterval, for the partition.

  • In particular, if is continous on [a,b], then 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 =
    where by we mean the average value of .

    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
    • #33
    • #21