On this display the area under the acceleration curve is calculated by breaking the time axis up into 100 intervals 0.1 second each, called "time slices". The area under the acceleration curve up to time=t is the sum of the areas of each time slice to the left of t. The area of a time slice is simply its base times its height. In this example the base of each time slice is 0.1 seconds. In general we will call the thickness of a time slice dt standing for delta t. The height of each time slice is 2 since we are dealing here with a constant acceleration of 2 m/s/s.
The change in the magnitude of the velocity during each interval dt is the area of the time slice corresponding to that dt. The magnitude then at any time, t, of the velocity is its initial value, here zero, plus the sum of all the little changes during all the dt intervals up to time t. We can plot the curve of velocity as a function of time by adding the area of each time slice to the previous value of velocity.