Acceleration – Displacement Graph

In terms of calculus, How do i prove that the area under the acceleration time graph gives me the change in KE / mass…..

Attempt at solution….

Area under the graph of a function y = f(x) is given by ∫y dx

Therefore area under acceleration displacement graph is ∫a ds (considering the function to be a=f(s))

=> ∫f/m ds

=> 1/m (∫f.ds)

=> 1/m (W) ………[work done = ∫f.ds ]

=> ΔKE/m ……..[W = ΔKE]

1. Is this proof correct mathematically and theoretically….
2. Is there a different or better proof…

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