Q:  What is the mass of this little piece?

        mass of piece     =  (length) * (linear density)

                          =  (dx) * (lambda)

                          =   lambda * dx


  Q:  What is the contribution of this little piece
          to the equation for center of mass?

        (mass) * (position)   =  (lambda dx) * (x)

                              =    lambda * x dx
        


  Q:  What is the location of the center
        of mass of the ruler in the horizontal direction?


               integral over all pieces (  lambda * x dx  )
   X com  =   ----------------------------------------------------------
                         total mass


                               /
                           int |  ( lambda * x dx  )
                               /
   X com  =   ----------------------------------------------------------
                               /
                           int |  ( lambda * dx  )
                               /



                             (   1   2  ) x = L
                   lambda *  (  --- x   )
                             (   2      ) x = 0
              =   ----------------------------------
                             (          ) x = L
                   lambda *  (   x      )
                             (          ) x = 0


                                      2           2
                   lambda *  (   1/2 L    -  1/2 0    )
              =   --------------------------------------
                   lambda *  (     L      -     0     )


                                      2
                   lambda *  (   1/2 L  ) 
              =   --------------------------------------
                          L * lambda



                   1
              =   --- L 
                   2