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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.
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.
In statistics you can find the area under a curve to establish what to expect between two input numbers. If there is a lot of area under the curve the graph is tall and there is a higher probability of things occurring there than when the graph is low.