COALITION MATH 152, SPRING 1998

              (S. A. Fulling, assisted by Vera Rice)

                             Day 22.T


    1.  The usual paper shuffling:  Old homework, CAPA, revised 
    new homework.  The next lab (Euler's method) is in the 
    book; no handout. 

    2.  Review moments of inertia from the beginning; follow the 
    Web through the cylinder example and talk about integration 
    in polar coordinates in general.  Promise a more difficult 
    example later if time permits (point 5 below).  Remark that 
    moment-of-inertia calculations for rectangular objects (in 
    rectangular coordinates) are not as hard as the wording of 
    the Web page seems to imply.

    3.  Laminas 

        A.  Follow Web through the theory, but not the example.

        B.  TEAM EXERCISE (an application of calculus to the 
        history of popular culture of the 1980s): 
        Find the centroid of a Pac-Man of mouth angle \pi/3
        (radius = 1, origin = vertex = center of circle).
        Odd teams:  rectangular coordinates (set up only)
        Even teams:  polar coordinates.

        Answers:  A = 5\pi/6
        A x-bar = - 1/3 => x-bar = - 2/(5\pi) = - .13...
        y-bar = 0 by symmetry

        C.  The rectangular calculation leads to the integral of
        \sqrt{1 - x^2}.  CHALLENGE:  Find this antiderivative.
        (There are two possible methods, one of which you have 
        been taught; the other is easy to find in the book.  I 
        exclude the method of finding the integral in the table 
        on the inside covers.)
THIS WAS STILL IN PROGRESS WHEN THE PERIOD ENDED.  EXERCISE 5 
WILL BE JETTISONED.

    4.  Terminological nuisances (following the Web).

        A.  "Moment" in first-degree and generic senses.

        B.  x-bar = M_y/M, etc.

    5.  TEAM EXERCISE:  Consider the pipe of CAPA problem 3.
    A pipe has inner radius 1 cm and outer radius 1.1 cm.  The 
    pipe is constructed of alternating layers of two materials.  
    The materials have diffused into each other, so that the 
    density is a continuous function of the radial coordinate, 
    \rho = A +  {\cos(B\pi r) \over r} \quad {\rm g/cm^3}.
    (A > 1 and B is in the range 20 ... 200.)  

        A.  Find the moment of inertia of a 50-cm length of this 
        piping (about the axis down the middle).

        B.  How would the answer change (setup only) if the ends 
        are beveled off (as by a pencil sharpener) at a 45 degree 
        angle?