% Calculating integrals
%
% Carlos R. Paiva
%
% 28 September 2021

close all
clear all

n_max = 1000;
x_min = 0;  x_max = 1;
y_min = 0;  y_max = 1;
delta = 0.25

for n = 1:n_max
    aux = (x_max-x_min)*(n-1)/(n_max-1)+x_min;
    x(n) = aux;
    y(n) = sqrt((1-aux)/(1+aux));
end

patch([0 x],[0 y],[0.8 0.8 1],'EdgeColor','none')
axis off
hold on
plot(x,y,'b','LineWidth',2)

X = x_max + delta;
Y = y_max + delta;
quiver(0,0,X,0,0,'r','LineWidth',2)
quiver(0,0,0,Y,0,'r','LineWidth',2)

p = 8;
plot(0,0,'ok','MarkerFaceColor','r','MarkerSize',p)
plot(x_max,0,'ok','MarkerFaceColor','r','MarkerSize',p)
plot(0,y_max,'ok','MarkerFaceColor','r','MarkerSize',p)

format long

analytical_integration = pi/2-1

f = @(x) sqrt((1-x)./(1+x));

tol = 1e-18;
numerical_integration = quad(f,0,1,tol)