COALITION MATH 152, SPRING 1998

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

                             Day 20.R


    1.  Review disk and cylinder methods (D, E, F of previous 
    day).

        A.  Implement on example x = 1 - y^2, rotated around each 
        axis.

        B.  Why is the volume of a thin cylindrical shell equal 
        to area times thickness?  Consider a thick cylinder.

        V = h\pi (difference between outer and inner R).

        For thin shell, use differentials:

        dV = 2\pi hR dR  (with R = R_1 \approx R_2).

        Equivalently, let R_2 = R_1 + dr, so

        R_2^2 - R_1^2 = R_1^2 + 2 R_1 dr + dr^2 - R_1^2
                \approx 2 R_1 dr   to first order, 

        so  V(shell) \approx 2\pi rh dr.

    2.  Washers:  a generalization of disks, sometimes needed.

    3.  TEAM EXERCISE (not collected):  #1, Sec. 5.2 (p. 324): 
    y = x^2, x = 1, y = 0, rotated about x axis.

        A.  Sketch the 2D region, the 3D region, and a typical 
        disk or washer. 

        B.  Find the volume.

        C.  Find the percentage error in the answer AutoCad gave 
        you.

        D.  (Later do the same for #9.)

    4.  TEAM EXERCISE (collected):  Based on #36, Sec. 5.3 (p. 
    330):  Consider 2D region bounded by y = x\sqrt{1+x^3},
    y = 0, x = 0, and x = 2.  "Set up" the integral, and evaluate 
    it if possible.

        A.  Revolve about x axis.

            i.  Disks/washers:  Teams = 1 mod 4

           ii.  Cyl. shells:  Teams = 2 mod 4

        B.  Revolve about y axis.

            i.  Disks/washers:  Teams = 3 mod 4

           ii.  Cyl. shells:  Teams = 4 mod 4
STUDENTS HAD AMPLE TIME TO WORK ON THIS, BUT NOT TO REPORT.  IT 
IS THE FIRST ORDER OF BUSINESS FOR NEXT DAY.