A bar magnet M falling under gravity through an air cored coil C . plot a graph show variation of induced emf (E) with time(t). what does the area enclosed by the E-T curve depict ?

We know that

induced emf is directly proportional to the rate of change of flux. Now, as the magnet falls the rate of change of emf linked

 with the coil increases steadily, until it reaches a maximum positive value and after that it starts to decreases as the rate of change of flux decreases until it becomes zero. After zero the rate of change of emf beings to rise and thus E starts rising as well until reaching a maximum and steadily decreasing down to zero.

The E vs t plot results.

Now we know that

E ∝ -dε/dt 

so,

E.dt ∝ -dε

or

E.t ∝ -ε

thus, the area under the E vs. t curve shows the magnetic flux linked with the coil.