% :::::::::::::::::: Integral mit Trapez-  &  Quad -  Function  :::::::::::::::::::::::
% ::::::::::::::::::::::::::  :::::::::::::
clear all ; close all  ; clf;
disp(['°°°°°°°°°°°° S T A R T °°°°°°°°°°°°°°°°°°°° ',date,' °°°°°°°°°°°°°°°°°°°°°'])
format compact
% ::::::::::::::::::::::::::::.
%% EF 230 Matlab Numerical integration example
clear all, close all, clc, 
format compact, format long g; 

%% Numerical integration of a function from x1 to x2
F = inline('sin(t)');               % function to integrate
x1 = 0;
x2 = pi;                                  % limits: x1 & x2
a1 = quad(F,x1,x2);                 % use quad to integrate

%% Show the curve and display a message to define the problem
fplot(F,[x1,x2])                            % a quick way to plot a function
msg = sprintf('What is the integral of %s from %.2f to %.2f?',func2str(F),x1,x2);
disp(msg);
waitfor(msgbox(msg));
%% Show the result
msg = ['Area calculated by the quad function = ' num2str(a1,10)];
disp(msg);
waitfor(msgbox(msg));
%% Approximate the integral via trapz for different numbers of points
for np = [5 10 25 50]
   clf                                                   % clear the current figure
   hold on                                            % allow stuff to be added to this plot
   x = linspace(x1,x2,np);                      % generate x values
   y = F(x);                                           % generate y values
   a2 = trapz(x,y);                                 % use trapz to integrate
   % Generate and display the trapezoids used by trapz
   for ii = 1:length(x)-1
       px = [x(ii) x(ii+1) x(ii+1) x(ii)];
       py = [0 0 y(ii+1) y(ii)];
       fill(px,py,ii)
   end
   fplot(F,[x1,x2]);                                   % plot the actual curve for reference
   msg = sprintf('Area calculated by trapz function with %u points = %.8f',np,a2);
   disp(msg);
   waitfor(msgbox(msg));
end