clear all     % clear all variables
close all     % closes figures between each run
clc           % clear command window between each run
format short  % to see only 5 digits
% format long % uncomment to see manny digits 

x = 0:pi/4:pi;      % [rad] 1x5 vector
% Type size(x) in Command window or check size of vecs in Workspace window
y = sin(x);          % 1x5 vector, x input needs to be in [rad].

figure(2)
plot(x,y,'-bx'), hold on
xlabel('x [rad]')
ylabel('y=sin(x)')

s=x(2)-x(1); % step size, between points

%% Area as typed into calculator, simpson vs trapez
As_calc= s/3*(y(1) + 4*y(2) + 2*y(3) + 4*y(4) + y(5)) % y(ind) takes index from vector y
At_calc= s/2*(y(1) + 2*y(2) + 2*y(3) + 2*y(4) + y(5)) % y(ind) takes index from vector y

%% Area calculated using vector multiplication of factors with y-values
sf= [1 4 2 4 1]; % Simpson factor vector
trf=[1 2 2 2 1]; % Trapez factor vector 

As=s/3 *  (sf*y') % Simpson rule, vector multiplication, y' transform necessary 1x5*5x1=1x1
% alternatively:
%As=s/3 * sum(sf.*y) % Simpson rule, element-wise multiplication using .*

At=s/2 * ( trf*y' ) % Trapez rule comparison

%% Area calculated using built-in trapz and HiB developed simps function
% Automatically adjusts length of sf and trf for different x, y input

As_hib_fun=simps(x,y)      % HiB developed function, on It's Learning
At_matlab_fun = trapz(x,y) % Matlab Trapez built-in function 
% Parenthesis () is used after all built-in Matlab functions, type >>help trapz in Command window to see what input is needed

%% Exact solution should be A = 2; decrease step size and trapz and simps
% should move towards this value.
x_fine = 0:pi/200:pi;
y_fine = sin(x_fine);
As_fine = simps(x_fine,y_fine)
At_fine = trapz(x_fine,y_fine)

%% Above can be used to calculate the area under any points y(x) even when 
% the function f(x) is not known. For a known function you can also define the
% function and take the exact integral: 
F = @(x)sin(x);
I_exact = integral(F,0,pi)
disp('Conclusion: Simpson closer to exact result A=2 when using few points.')

%% Example 2 on page 33-34 in Rawson&Tupper is a good way to repeat different
% integration methods at home.