also Morera's theorem states that f should be continuos and int(f) = 0. Does the above e^xt --> 0 show int(f) = 0, or do we have to solve the integral?
i suppose f is not discontinuous on S?
Does this show that the integral has a finite limit?
No, but the fact that $\displaystyle e^{(-ct)}$ decays faster than any polynomial, for any $\displaystyle c>0$ does. For the last one use Green's theorem, Fubini and the representation of $\displaystyle F'$ you got.