>In fact, there are cases where interchanging the order is not allowed when you're in that situation depending on the integrand.
yup exactly in the cases where fubini's theorem doesn't hold and therefore those cases for which the integral theorem doesn't hold.
Leibniz Theorem: Let f(x, t) be a function such that both f(x, t) and its partial derivative fx(x, t) are continuous in t and x in some region of the (x, t)-plane, including a(x) ≤ t ≤ b(x), x0 ≤ x ≤ x1. Also suppose that the functions a(x) and b(x) are both continuous and both have continuous derivatives for x0 ≤ x ≤ x1...
Fubini's Theorem: If f(x,y) is a *continuous function* on a rectangle R=[a,b]×[c,d]...