Day 04 in Mat122

Today:

• Review of Section 5.1: approximating and computing area, and extensions
  • We found four different estimates for the following problem:

    Suppose that you clock your speed at each minute along a trip, in mph:

    minute	0	1	2	3	4	5	6	7	8	9	10
    speed (mph)	0	30	45	30	70	65	70	70	45	30	35

    Estimate the distance travelled.

  • Estimates:
    • Bogdan's rule: d=44.55*10 = 7.42 miles
    • RRR: d = 45.5*10 mph*minutes = 7.58 miles
    • LRR: d = 49*10 mph*minutes = 8.17 miles
    • Trapezoidal rule = 47.25*10 mph*minutes = 7.88 miles

      Notes:

      • Whenever you have two estimates, you have a third:
      • the trapezoidal rule is the average of left and right rectangle rules.

  • Each of these estimates corresponds to computing a rectangle height (an average speed), and then multiplying by the total time (10 minutes). Hence we're using the "dirt" formula, and computing the average (the "area under the curve", even though there's no curve!) in several different ways.

  • When you have a curve (or a formula) for f, you can take the time intervals shorter and shorter, until you get as close to the area as you need: (notice that : that is, that )
    Then, in fact,

  • Examples:
    • #8, p. 258
    • #9, p. 258 (also trapezoidal)

  • Generalizations?
    • The area can be calculated by finding an average value on the interval, times the length of the interval: Area under the graph of f from a to b = where by we mean an average value of f.

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

  • Questions?

• Section 5.2: The definite integral
  • This section formalizes our intuition from the preceding section:
    • First of all, we can't expect our function to be positive all the time, so we define "signed area":
      

      and then we give a name to this area:

      

    • 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.
        

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

        and where C is a set of intermediate points C={c₁,...c_N} such that

      • 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.

    • Here's how we'd rewrite our generalization that Area under the graph of f from a to b = : where by we mean an average value of f.

      This is a preview of section 6.2, p. 315 -- the Mean Value Theorem for Integrals.