# Is "Gradient Computation" in Finite Volume Discretization Really 2nd order accurate?

Based on this, pp 245, we go through these steps to discretize a gradient statement, namely $$\nabla\phi$$:
$$\int_V\nabla \phi dV = \oint_{\partial V}\phi dS$$ 2- Integral mean value theorem, applying on one cell (i.e. $$c$$) results, $${\bar{\nabla\phi}}_c =\frac 1 {V_c} \oint_{\partial V_c}\phi dS$$ 3- Applying $$2^{nd}$$ order accurate discretisation for RHS, we end up with ($$f$$ stands for cell face centroids), $${\bar{\nabla\phi}}_c =\frac 1 {V_c}\Sigma \phi_f S_f$$ The question is :
Does applying Integral mean value theorem, namely $${\bar{\nabla\phi}}_c =\frac 1 {V_c} \int_{V_c}\nabla \phi dV$$, maintain $$2^{nd}$$ order accuracy of discretisation?