date 
 unsetenv DISPLAY 
 cd ~/Symbolic/Maple/ 
 maple
pollux: pollux: Sun Nov 12 18:05:50 EST 1995
pollux: pollux: {{Maple}}     |\^/|     Maple V Release 3 (University of Florida)
._|\|   |/|_. Copyright (c) 1981-1994 by Waterloo Maple Software and the
 \  MAPLE  /  University of Waterloo. All rights reserved. Maple and Maple V
 <____ ____>  are registered trademarks of Waterloo Maple Software.
      |       Type ? for help.
                                     20872


# Part e  of the first problem on TakeHome B.

> int( y*2* sqrt(1-y^2) , y=0..1) ; 

                                      2/3

# The quote symbol, " ,below means `the previously printed value'.

> ymoment := " ;

                                 ymoment := 2/3

> area := (1/2)*Pi*1^2;

                                 area := 1/2 Pi

> ycentroid := ymoment / area ;

                                              4
                               ycentroid := ----
                                            3 Pi


> # Consider the big disk and small disk, of radius b and s respectively.

> ycen_b := b*ycentroid ;


                                               b
                               ycen_b := 4/3 ----
                                              Pi

> xcen_b := 0 ;

                                  xcen_b := 0

> ycen_s := s*ycentroid ;

                                               s
                               ycen_s := 4/3 ----
                                              Pi


# Center of the small disk is translated by b-s from the origin.

> translation := b - s ;

                              translation := b - s

> xcen_s := 0 + translation ;

                                xcen_s := b - s

> weight_b := b^2*area ;

                                              2
                             weight_b := 1/2 b  Pi

> weight_s := -(s^2*area) ; # NEGATIVE, because we are REMOVING the small disk.

                                               2
                            weight_s := - 1/2 s  Pi


# Now compute the location of the centroid for the green figure.  Let's
# remind ourselves of the value we have computed so far:


> xcen_b; ycen_b; xcen_s; ycen_s; 

                                       0

                                          b
                                    4/3 ----
                                         Pi

                                     b - s

                                          s
                                    4/3 ----
                                         Pi

> weight_b ; weight_s;

                                        2
                                   1/2 b  Pi

                                         2
                                  - 1/2 s  Pi


> xmoment_both  := (weight_b * xcen_b) + (weight_s * xcen_s) ;

                                             2
                      xmoment_both := - 1/2 s  Pi (b - s)


>  xcen_both  := xmoment_both/(weight_b  + weight_s) ;

                                            2
                                           s  Pi (b - s)
                    xcen_both := - 1/2 ---------------------
                                            2           2
                                       1/2 b  Pi - 1/2 s  Pi


> ymoment_both  := (weight_b * ycen_b) + (weight_s * ycen_s) ;

                                             3        3
                        ymoment_both := 2/3 b  - 2/3 s


>  ycen_both  := ymoment_both/(weight_b  + weight_s) ;

                                            3        3
                                       2/3 b  - 2/3 s
                       ycen_both := ---------------------
                                         2           2
                                    1/2 b  Pi - 1/2 s  Pi


> dist_squared  := xcen_both^2 + ycen_both^2 ;

                              4   2        2                3        3 2
                             s  Pi  (b - s)           (2/3 b  - 2/3 s )
    dist_squared := 1/4 ------------------------ + ------------------------
                              2           2    2         2           2    2
                        (1/2 b  Pi - 1/2 s  Pi)    (1/2 b  Pi - 1/2 s  Pi)


> dist_squared := simplify(") ;

                          4         3       2  2       3        4   2       4
                      16 b  + 32 s b  + 48 s  b  + 32 s  b + 9 s  Pi  + 16 s
  dist_squared := 1/9 -------------------------------------------------------
                                                   2   2
                                            (b + s)  Pi

# Now we plug in the numbers from the problem.

> b := 1 ; s := 1/2 ;

                                     b := 1

                                    s := 1/2

> dist_squared ;

                                                2
                                    49 + 9/16 Pi
                               4/81 -------------
                                           2
                                         Pi


# Now I check the answer on Chris,Stacey,John's paper.

> chrstajoh := (9*Pi^2 + 784)/ (18*Pi)^2 ;

                                                2
                                            9 Pi  + 784
                         chrstajoh := 1/324 -----------
                                                  2
                                                Pi


> evalb(simplify(dist_squared = chrstajoh)) ;

                                      true

# It is correct.  



# Now I check the answer on Adrienne and Byron's paper.

> aby := ((1/36) + (196/(81*Pi^2))) ;

                                              196
                              aby := 1/36 + ------
                                                 2
                                            81 Pi

> evalb(simplify(dist_squared = aby)) ;

                                      true


# It is correct.