COALITION MATH 151, FALL 1997

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

                        Days 11.2 and 11.3


FIRST FINISHED PREVIOUS DAY (PARTS 2 AND 3).
1.  Arc length; one-dimensional densities

    A.  Today, and frequently from now on, we will operate in 
    "setup" mode.  To SET UP an integral means 

        i.  DON'T try to evaluate it.

       ii.  DO make the integrand and the limits of integration 
        explicit (i.e., NOT "\int_a^b f(x) - g(x)").

    B.  Activity:  Find the length of the curve

        x = t^3,  y = t^4,  0 < t < 1  [Stewart 9.3.1, p. 563].
 
IN REACTION TO PRIVATE STUDENT QUESTIONS, I POINTED OUT THAT WHEN 
C IS THE GRAPH OF A FUNCTION, ds = \sqrt{1 + (f')^2} dx.  (SOME 
STUDENTS HAVE SEEN THIS FORMULA BEFORE AND ARE MYSTIFIED BY THE 
ABSENCE OF A "+ 1" IN THE GENERAL ARC LENGTH FORMULA.)  I ASKED 
THE STUDENT WHO FIRST ASKED THE QUESTION TO EXPLAIN THE MATTER TO 
THE CLASS.
     
    C.  Digression on density

        i.  Activity:  Find the cost of a highway 5 miles long, 
        one mile of which is a causeway across a bayou costing 
        $1000/foot.  (The rest is ordinary pavement, $200/ft.)

       ii.  Activity:  Find the cost of a highway 5 miles long 
        that runs into an increasingly populated area.  As the 
        traffic increases, it is necessary to make the highway 
        out of more durable (and expensive) material.  At point x 
        (0 < x < 5), the cost of the road is $200 + x per foot.

      iii.  Remarks

        a.  We have two natural interpretations of a definite 
        integral:

            (1) area of a 2-dim region

            (2) total value of a quantity associated with a 1-dim 
            region, accumulated from a value (density) attached 
            to each point.  (Work is another example.)

        b.  We met densities before in the section on "rates of 
        change in the natural and social sciences".  Density =

            (1) amount of stuff in an interval, divided by length 
            of interval.
    
            (2) derivative of the amount of stuff that exists 
            from the beginning to point x.   
    
        Now that we know the fundamental theorems of calculus, it 
        should be much clearer what is going on here!  A density 
        is something whose destiny is to be integrated, and with 
        that end in mind it is defined as some sort of 
        derivative.

        c.  All this might seem more intuitive if we were working 
        with 3-dimensional regions.  E.g., density ordinarily has 
        units of mass per unit volume (kg/m^3); so far we can 
        deal only with density of a wire, mass per unit length 
        (kg/m).  Next week we will graduate to 2 dimensions and 
        can talk about the density of a plate!

GOT THIS FAR.  ACTIVITIES SO FAR WERE DONE WITHOUT BENEFIT OF 
TRANSPARENCIES, WITH ONLY THE REPORTING TEAM GETTING CREDIT.  ON 
THE NEXT DAY THE FOLLOWING EXERCISE WILL BE DONE WITH 
TRANSPARENCIES, AS A SEMI-RAT.

DAY 11.3 WENT VERY SLOWLY, PARTLY BECAUSE THE OVERHEAD PROJECTOR 
WAS BROKEN, PARTLY BECAUSE THE STUDENTS WERE CLEARLY NOT YET 
COMPETENT IN SETTING UP AND EVALUATING THESE INTEGRALS.

    D.  Integrals with respect to arc length

        i.  Activity:  The highway runs up a mountain with 
        equation  x = t,  y = \sqrt{t + 1}/10  (0 < t < 5).  
        Because of erosion problems, higher altitude requires 
        more durable material, so the cost at point x is $200 + y 
        per foot. Set up an integral for the cost of the highway.  
        (DON'T neglect effect of slope on length.) 

       ii.  Such integrals can also have an area interpretation.

            a.  highway of variable width

            b.  wall above C of variable height (Stewart drawing, 
            p. 897).

      iii.  Summary:  There are two kinds of "line integrals":

            a.  Vector field as integrand, \int_C F(r).dr  
            (e.g., work) 

            b.  Scalar density as integrand, \int_C \rho(s) ds
            (e.g., mass or cost of a one-dimensional object of 
            varying composition; concrete in a wall of varying 
            height)

2.  Path independence of line integrals

    A.  Activity:  Find \int_C F.dr for F = <2x sin y, x^2 cos y>

        i.  Odd teams:  C = straight line from origin to (1,1).

       ii.  Even teams:  C = L-path from origin to (1,0) and 
       thence to (1,1).
GOT THIS FAR ON DAY 11.3; MUST STILL DO i AND REVIEW ii ON DAY 
12.1!  THIS EXAMPLE IS DEFECTIVE FOR THE PURPOSE INTENDED, 
BECAUSE METHOD i LEADS TO INTEGRALS THAT REQUIRE INTEGRATION BY 
PARTS, SO STUDENTS MUST TAKE MY WORD FOR IT THAT THE ANSWER IS 
THE SAME AS BY THE OTHER METHOD.

    B.  Are line integrals ALWAYS independent of path?  Recall 
    the example of F = <-y,x> integrated around a circle.

    C.  Integral always independent of path <=> integral around a 
    CLOSED path is always 0.

3.  Gradients, potential energy, and all that [tomorrow]
    
4.  Double integrals and Green's theorem  [next week]     
TWO WEEKS LATER, AS IT TURNED OUT!