% Example usage: 
%  x = 0:.001:1;
%  f = exp(x) + exp(-x);
%  dfdx = exp(x) - exp(-x);
%  tic,fhat = intgrad1(dfdx,.001,2,2);toc

 
% psuedo code
f = plot(x, y);
A = integrate(f);

 
% intgrad: generates a vector, integrating derivative information.
% usage: fhat = intgrad1(dfdx)
% usage: fhat = intgrad1(dfdx,dx)
% usage: fhat = intgrad1(dfdx,dx,f1)
% usage: fhat = intgrad1(dfdx,dx,f1,method)
%
% arguments: (input)
%  dfdx - vector of length nx, as gradient would have produced.
%
%    dx - (OPTIONAL) scalar or vector - denotes the spacing in x
%         if dx is a scalar, then spacing in x (the column index
%         of fx and fy) will be assumed to be constant = dx.
%         if dx is a vector, it denotes the actual coordinates
%         of the points in x (i.e., the column dimension of fx
%         and fy.) length(dx) == nx
%
%         DEFAULT: dx = 1
%
%    f1 - (OPTIONAL) scalar - defines the first eleemnt of fhat
%         after integration. This is just the constant of integration.
%
%         DEFAULT: f1 = 0
%
%  method - (OPTIONAL) scalar - either 0, 1, 2, or 3. Defines
%         the integration scheme used.
%
%         method = 0 --> cumtrapz
%
%         method = 1 --> solves central finite difference
%                        approximation using linear algebra
%                        A second order fda. At least 3 points
%                        are necessary.
%
%         method = 2 --> integrated spline model
%                        This will almost always be the most
%                        accurate among the alternative methods.
%
%         method = 3 --> integrated pchip model
%
%         method = 4 --> higher order finite difference approximation
%                        A 4th order fda. At least 5 points are
%                        necessary.
%
%         DEFAULT: method = 2
%
%         Note: method = 0 (cumtrapz) will generally be the fastest,
%         and method = 2 (spline integral) will be the most accurate
%         of the four methods.
%         Methods 1, 3, and 4 were put in there mainly for fun on my
%         part, lthough for equally spaced points, the 4th order fda
%         should also be quite accurate.
%
%         Data series with noise in them may be best integrated using
%         a lower order method to avoid noise amplification.
%
% arguments: (output)
%   fhat - vector of length nx, containing the integrated function
%
% Example usage: 
%  x = 0:.001:1;
%  f = exp(x) + exp(-x);
%  dfdx = exp(x) - exp(-x);
%  tic,fhat = intgrad1(dfdx,.001,2,2);toc

function I = jrbTrap(x,y)
Y = (y(2:end)+y(1:end-1))/2;
X = x(2:end)-x(1:end-1);
I = sum(X.*Y);

EDU>> x = sort([0 rand(1,9)*10 10]);
EDU>> y = x.^2;
EDU>> I = jrbTrap(x,y)

I =

  337.3206

EDU>>