The area under the velocity-time graph between any two instants t = t₁ and t = t₂ gives the displacement covered in time $\delta$t = t₂- t₁:

only if the particle moves with a uniform velocity.

only if the particle moves with a uniform acceleration.

only if the particle moves with an acceleration increasing at a uniform rate. 

in all cases irrespective of whether the motion is one of uniform velocity, or of uniform acceleration or of variable acceleration.