If $F(s) = \mathcal{L}\left\{ f(t) \right\}$, then $\displaystyle \int_0^\infty e^{-st} f(t) \, dt = F(s)$.
Taking the limit as $s \to 0$, then $\displaystyle \int_0^\infty f(t) \, dt = F(0)$ assuming the integral to be convergent.
If $F(s) = \mathcal{L}\left\{ f(t) \right\}$, then $\displaystyle \int_0^\infty e^{-st} f(t) \, dt = F(s)$.
Taking the limit as $s \to 0$, then $\displaystyle \int_0^\infty f(t) \, dt = F(0)$ assuming the integral to be convergent.