function box

syms x;
n=100;                          %initial number of subintervals
f=exp(-x^2);                    %function to be integrated over (0,1)
exact=eval(int(f,0,1));         %exact integral
fprintf('exact=%.10f \n',eval(int(f,0,1)));


for i=1:15                       
    m=n*2^(i-1);                %divide each interval into two pieces of subintervals
    t=0:1/m:1;
    box=subs(f,t);              %evaluate the fucntion at each grid
    boxsum(i)=sum(box)/m;       %sum of the areas of boxes=appoximate integral
end
error=abs(exact-boxsum);        %error 

for i=1:14
    ratio(i)=error(i)/error(i+1);   
end

fprintf('\n approximation \n')
fprintf('%.10f \n',boxsum)

fprintf('\n error \n')
fprintf('%.10f \n',error)

fprintf('\n ratio \n')
fprintf('%.10f \n',ratio)