Maxwell's area construction, why? In the standard derivation of Maxwell's area construction (which can be found on page 4 of this pdf) the following equation is often written:
$$G(p_1,T)=G(p_0,T)+\int^{p_1}_{p_0}Vdp $$
When the two phases are in equilibrium  $G(p_1,T)=G(p_0,T)$ so the integral must vanish. From this it is said that Maxwell's area construction must hold. Please can someone explain this to me? For the integral should vanish for any two points on the same isotherm at the same pressure, and therefore following the standard argument, Maxwell's area construction should work for any two such points?
 A: Maybe I haven't read correctly the pdf you put as a link but it does not write exactly what you write. 
That being said, let me try to explain the Maxwell's construction without appealing to the Gibbs free energy function.
To this, I just consider a van der Waals fluid with the various isotherm PV curves that you have in the first pages of the pdf you put as a link.
If you were to integrate this pressure curve to get the corresponding Helmholtz free energy $A(N,V,T)$ you would get the black curve in the figure below

What we can see is that after point 1, the free energy displays a local concavity in its graph up to point 2. If we follow the rule that a thermodynamic system always follows a strategy that minimises free energy, it so happens that one way to do it is to take the convex hull of the van der Waals free energy curve (in red) where the concave part is replaced by a straight segment linking points 1 and 2.
Proposing that a real system would prefer following a straight segment from 1 to 2 rather than a locally concave part implies that the thermodynamic pressure in that region (which turns out to be the coexistence region) has to be constant.
The role of the Maxwell's construction is simply to ensure that upon replacing the van der Waals pressure curve in the coexistence region of the PV diagram by a flat horizontal segment, we still bridge points 1 and 2 of the initial van der Waals free energy curve depicted above i.e. the extra gain and loss in free energy, relative to the convex hull, compensate. It just so happens that, additionally, this constraint is equivalent to asking the chemical potential to be the same in the two coexisting phases.
A: Maxwell construction is about a system transitioning from one phase to another phase but not about two phases in equilibrium. When changing from one phase to another phase, there is a path that pressure and volume can change. However, because the Gibbs energy of the change is higher than the simple phase change, the pressure and volume make abrupt change from phase 1 to phase 2. Maxwell construction tells us how to calculate this abrupt change. 
