The mathematical technique used to calculate the area under a graph is integral calculus. This is a complex subject which I am not going to attempt to explain in detail- it normally takes years of study.
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The area under an acceleration-time graph is equal to the object's velocity (not change in velocity).
It is not, if it is a graph of force against acceleration.
change in velocity
Displacement is the area under the v-t graph.
The distance travelled over the time period represented by the area under the v-t graph between the end points.