clear
format long

%%
% We use the trapezoidal integration formula to compute the perimeter of an
% ellipse with semi-axes 1 and 1/2. Parameterizing the ellipse as x = cos(pi t), 
% y = 1/2 sin(pi t) leads to the integral
%
% \int_{-1}^1 \pi\sqrt{ \sin^2(\pi t) + \tfrac{1}{4} \cos^2(\pi t)} dt
%
%%
f = @(t) pi*sqrt( sin(pi*t).^2 + cos(pi*t).^2/4 );
N = (4:4:60)';
C = zeros(size(N));
for i = 1:length(N)
    h = 2/N(i);
    t = h*(0:N(i) - 1);
    C(i) = h*sum(f(t));
end

perimeter = C(end)
err = abs(C - C(end));
semilogy(N, err, '.-')
title('Convergence of perimeter calculation')
xlabel('number of nodes')
ylabel('error')
grid on
    
%%
% The approximations gain about one digit of accuracy for each constant
% increase in N, consistent with geometric (linear) convergence.