COALITION MATH 151, FALL 1997

        Group II (S. A. Fulling, assisted by Cary Lasher)

                              Day 8.2


THIS LECTURE WAS GIVEN BY CARY WHILE I WAS OUT OF TOWN.  HE 
PROBABLY DID NOT FOLLOW THIS SCRIPT EXACTLY.

Area and Riemann sums:  To motivate the definition of a DEFINITE 
INTEGRAL we consider two prototype application problems: 

    1. Area under a curve y = f(x) > 0 between x = a and x = b.

        A.  Preliminary motivation based on the map of Texas.  
        (Should distinguish between area on the map (graph) and 
        area on the ground.  Cf. slope ambiguity in first 
        integrated exam.)
GOT THIS FAR ON DAY 8.1.

        B.  Cut region into narrow vertical strips.  (Keep widths 
        equal for now.)

        C.  Whatever we mean by the area of a strip, it is 
        certainly trapped between the rectangles with height 
        equal to the max and the min of f(x) over that strip.

        D.  If the function is monotonic, the max will always be 
        at one end of the strip and the min at the other.  (There 
        is not much loss of generality here, since any function 
        we expect to meet on the street will have only finitely 
        many extrema between a and b.) 

        E.  Therefore, the area of the region is trapped between 
        the "left sum" and the "right sum".  [Run the Maple demo 
        here.]

        F.  We can hope that as strip width goes to 0, the upper 
        and lower sums will meet at a number A, the true area (by 
        definition).  Examples:  Use i^n formulas from section on 
        summation notation.
                
    2. Finding the distance traveled by a body from its speed 
    history  (a foretaste of the "fundamental theorem of 
    calculus" -- recall the speedometer/odometer icon).

        A.  Constant speed implies distance = speed x time. 

        B.  Variable speed:  Above is still approximately true 
        on very short time intervals.  Therefore, cut time axis 
        into small pieces and add up speed x time to get total 
        distance.  Expect this to be exact as interval size goes 
        to 0.

        C.  Comparing with the area problem, we see:
        THE DISTANCE TRAVELED BETWEEN TIMES a AND b IS THE AREA 
        UNDER THE GRAPH OF THE SPEED FUNCTION (IN APPROPRIATE 
        UNITS).

        D.  But, in a previous week, we learned:
        THE DISTANCE FUNCTION IS AN ANTIDERIVATIVE OF THE SPEED 
        FUNCTION.

        E.  Conclusion:  Areas and antiderivatives are almost 
        the same thing!  Precise statement of this is the 
        "fundamental theorem of calculus", which we reach next 
        week.

    3.  But first we need a more general definition of the 
    operation of "chopping into strips and adding up":  Riemann's 
    definition of the definite integral. 

        A.  Riemann sums:  For technical reasons we need to allow 
        strip widths to vary and evaluation point in each strip 
        to be arbitrary.  Since we aren't going to study the 
        proofs, you won't suffer much if you just think in terms 
        of left, right, and midpoint sums with strip width = (b-
        a)/2^N.  (If time permits, mention a crazy function for 
        which these things do matter: characteristic function of 
        the rationals.) 

        B.  Extend definition to b < a and b = a.

        C.  Drive point home:  Areas and distances are expressed 
        as definite integrals.

        D.  Discuss "dummy variables": Integral is a number, not 
        a function; etc. 

        E.  If graph drops below the line, integral is still 
        defined but area interpretation must be modified.

        F.  The limit in the definition certainly exists if f is 
        continuous, and for lots worse functions than that.  If 
        function goes to infinity or the interval is infinite, 
        definition will require a second limit ("improper 
        integrals", next semester).