Motional emf through a non-uniform field

Suppose we have a rod of length L (in the upwards direction – y) moving to the right with velocity v. The magnetic field is into the plane of the page. If the field were constant the induced emf would be
vBL. Now suppose that the magnetic field
is varying in a 1/y^2 and 1/x^2 fashion. Does that mean we would have to use a double integral(first because B varies through the surface that lies in the y -axis and secondly because the rod is moving to the right (x-axis) where B is decreasing)? Secondly, if the rod was moving through time varying field would there would be no way to describe the magnetic field in the rod and therefore no way to find the induced emf is that right ?