Area under weird curve

%% Parametrization of curve
x = @(t) sin(t) + cos(t).*(pi-t);
y = @(t) -cos(t)+sin(t).*(pi-t);

thetas = linspace(0, pi, 10000);

curve.xi = x(thetas);
curve.yi = y(thetas);

circle.xi = sin(thetas);
circle.yi = cos(thetas);

%% Nice plot
figure(1)
clf
grid
hold on
plot(curve.xi, curve.yi, 'b.')
plot(circle.xi, circle.yi, 'y.')

%% Calculate area below blue curve
integral = 0;
for i = 2:numel(curve.yi)
    delta = abs(curve.xi(i)-curve.xi(i-1));
    integral = integral + delta*curve.yi(i);
end

% Total should be:
total = 2*(integral-pi/2) + pi^3/2;
disp(['Total = ' num2str(total)])