The last equation in the figure at the left states that the integral of E around any closed path is determined only by the time rate of change of magnetic flux passing through any surface bounded by that path.
The common cause of failure to use Faraday's law reliably is failure to relate the line integral and the surface integral. Trying to use Faraday's law by using only the top equation in the figure at the left is to ignore cause and effect.
Ignoring cause and effect is the basic cause of the phenomena popularly called misconceptions . The simple example is Newton's law, ma = F. The novice will often make statements about acceleration which do not reflect the cause, namely the force acting on an object.
The Faraday disk.