%%
clc;
clear;
%Q.) 7
%Evaluate integral of exp(-x)/sqrt(x) over [0,1].
%(a) Using a rectangular rule
%(b) Make a change of variables x = t^2 and use rectangular rule on new variable.
%After applying change of variables we get the integral to find out equal
%to integral of exp(-x^2) over [-1,1].
syms f(x);
f(x) = exp(-x^2);
syms g(t);
g(t) = exp(-t)/t^(1/2);
a = -1;
b = 1;
c = 0;
N = [5 10 20 50 100 200 500 1000];
exact_int = 1.4936482656248540507989; % exact value of integral evauated from wolfram alpha.
Ngrids = 8; % Number of different grids
h1 = zeros(Ngrids,1);
h2 = zeros(Ngrids,1);
rect_1 = zeros(Ngrids,1);
rect_2 = zeros(Ngrids,1);
for i = 1 : Ngrids
h1(i)=(b-a)/N(i); % Different-different grid spacing
h2(i)=(b-c)/N(i);
x = linspace(a,b,N(i)+1); % Grid Points
t = linspace(c,b,N(i)+1); % Grid Points
y1 = 0.5*(x(1:N(i))+x(2:N(i)+1)); % Midpoints of each panel in x(i)'s
y2 = 0.5*(t(1:N(i))+t(2:N(i)+1)); % Midpoints of each panel in t(i)'s
rect_1(i)=h1(i)*sum(f(y1)); % Rectangular rule after applying change of variables
rect_2(i)=h2(i)*sum(g(y2)); % Rectangular rule
end
error_rect_1 = abs(double(rect_1 - exact_int)); % absolute error using Rectangular rule after applying change of variables
error_rect_2 = abs(double(rect_2 - exact_int)); % absolute error using Rectangular rule
figure;
loglog(h1,error_rect_1,'r','linewidth',1);
hold on
loglog(h2,error_rect_2,'b','linewidth',1);
hold on
loglog(h1,h1.^2,'k');
hold on
loglog(h2,h2.^2,'g');
xlabel('Gridspacing');
ylabel('Error');
title('"Error vs GridSpacing" for Rectangular Rules with and Without changing of variables','FontSize',14,...
'FontWeight','bold','Color','k');
legend('Recangular rule after applying change of variables' ,'Rectangular rule','h1^2','h2^2','location','southeast','FontSize',11,...
'FontWeight','bold','Color','y');
