What is the proof that an area under a curve is the definite

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Calculus: Why does integrating a function give area under its curve?

Integration: The Area Under the Curve

What is the proof that an area under a curve is the definite integral of that function? - Quora

The limit of the lower and upper Riemann sums are equal which is what allows one to define area under a curve as the limit of a Riemann sum. Proof? - Quora

What's the real world application of knowing the area between two curves on a graph? - Quora

If the differentiation of a function is the function's gradient, how can the inverse process of differentiation (integration) determine the area of a closed path? - Quora

What's the real world application of knowing the area between two curves on a graph? - Quora