In MATLAB (R2017b, online version) I found an issue during symbolic integration: try to calculate integral of f(t) = abs(sin(t)) . Due to the fact that f(t) is always positive, I expect that the integral in [a, b] is lower equal than the integral in [a, b'] if b < b'. But:
int( abs(sin(t)), t, 0, pi ) -> 2 % OK
int( abs(sin(t)), t, 0, 2*pi ) -> 2 % NO (should be 4)
In fact, if we plot the function that represent the integral from 0 to x, which should be monotonic, we found something different:
% for each value x(i) of x we will calculate integral from 0 to x(i)
x = 0 : pi/8 : 4*pi;
% actual computation
Z = zeros(length(x), 1); % create array
syms t; % create symbolic variable t
calculate_int = @(n) int(abs(sin(t)), t, 0, n); % integral function
for i = 1 : length(Z)
Z(i) = calculate_int(x(i));
end
% plot result
figure;
plot(x, Z);
results in this obviously non monotonic function:
Non symbolic integration gives no problem:
x = 0 : 0.01 : pi;
f = abs(sin(x));
value = sum(f * 0.01) % 2
x = 0 : 0.01 : 2 * pi;
f = abs(sin(x));
value = sum(f * 0.01) % 4
