function result = polygon_yy_2d ( n, x, y )

%*****************************************************************************80
%
%% polygon_yy_2d() integrates the function Y*Y over a polygon in 2D.
%
%  Integration region:
%
%    The polygon bounded by the points (X(1:N), Y(1:N)).
%
%  Formula:
%
%    INTEGRAL = (1/12) * SUM ( 1 <= I <= N )
%      - ( Y(I)**3 + Y(I)**2 * Y(I-1) + Y(I) * Y(I-1)**2 + Y(I-1)**3 )
%      * ( X(I) - X(I-1) )
%
%    where X(0) and Y(0) should be replaced by X(N) and Y(N).
%
%  Licensing:
%
%    This code is distributed under the MIT license. 
%
%  Modified:
%
%    27 November 2004
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    S F Bockman,
%    Generalizing the Formula for Areas of Polygons to Moments,
%    American Mathematical Society Monthly,
%    1989, pages 131-132.
%
%  Input:
%
%    integer N, the number of vertices of the polygon.
%    N should be at least 3 for a nonzero result.
%
%    real X(N), Y(N), the coordinates of the vertices
%    of the polygon.  These vertices should be given in
%    counter-clockwise order.
%
%  Output:
%
%    real RESULT, the value of the integral.
%
  result = 0.0;

  if ( n < 3 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'POLYGON_YY_2D - Warning!\n' );
    fprintf ( 1, '  The number of polygonal vertices must be \n');
    fprintf ( 1, '  at least 3, but the input polygon has N = %d\n', n );
    error ( 'POLYGON_YY_2D - Fatal error!' );
  end

  for i = 1 : n

    if ( i == 1 )
      im1 = n;
    else
      im1 = i - 1;
    end

    result = result - ( y(i)^3 + y(i)^2 * y(im1) + y(i) * y(im1)^2 ...
      + y(im1)^3 ) * ( x(i) - x(im1) );

  end 

  result = result / 12.0;

  return
end
