Consider a Riemannian manifold (M^{m}, g) whose volume is the same as the standard sphere (S^{m}, g_{round}). If p\!>\!\frac {m}{2} and \int _{M}\! \left \{ Rc\!-\!(m\!-\!1)g\right \}_{-}^{p} dv is sufficiently small, we show that the normalized Ricci flow initiated from (M^{m}, g) will exist immortally and converge to the standard sphere. The choice of p is optimal.