What is the area in Faraday's law if we have only a piece of metal moving in a magnetic field? If a piece of metal of length $l$ is moving with a speed $v$ in a region where there is a uniform magnetic field $B$ perpendicular to it, there will be a potential difference across its terminals equal to $lvB$ which is known as motional EMF.  This can be shown and understood in terms of magnetic and electric forces on the free charges in the metal.
How can one calculate such EMF from Faraday's law, 
$\displaystyle\mathcal{E} = \left|\frac{d\Phi_B}{dt}\right|$? 
(where $\Phi_B$ is the magnetic flux $\int \bf{B}\cdot d\bf{a}$)
(If $B$ is not changing, then the change in the magnetic flux must be due to change in an area, but the area of what? What are the boundaries of this area?)
 A: The general Faraday law of emf formulated with magnetic flux is meant mostly for closed circuits made of thin wire, which can be assigned area without problem. 
For other situations, magnetic flux may not have sense. Moving piece of metal is still subject to magnetic electromotive intensity, but it has to be calculated for any point of the metal as 
$$
\mathbf E^* = \mathbf v \times \mathbf B
$$
where $\mathbf v$ is the velocity of the metal element.
A: I think your question answers itself, indeed the area of what? Faraday's law is for a closed loop of wire, thus Faraday's law is inappropriate and we should look for an alternative, as you have done by considering the Lorentz force. If the metal were a closed loop of circumference $l$ then Faraday's law would be valid. The forces on the electrons from the top and the bottom would both act in the same direction but in opposite orientations around the loop and thus no EMF around. Also the force on the protons would be counter to that on the electrons and thus no overall force on the loop, thus we conclude that Faraday's Law saying that there would be no net EMF is correct.
