In acceleration time graph how speed finding by area enclosed under the graph . detail understanding

Acceleration is defined as rate of change of speed. Let a(t) is acceleration which is varying with time.

 

Then we have,

it is assumed in eqn.(1), Speed v also varying with time.

 

Eqn.(1) can be written as,

 

If change in speed is from v_i to v_f in interval of time between t_i and t_f ,

we write, 

extreme RHS of eqn.(3) is integration of time varying acceleration function within the time interval t_i and t_f .

Hence as shown in figure the change in speed can be calculated as area under the curve, if acceleration is plotted against time