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?

In order to find the area between two functions, should the definite integrals of both functions be equal to a positive value for that space? - Quora

Where can I read the proof that definite integrals allow you to calculate the area under a curve? - Quora

Lesson Explainer: Properties of Definite Integrals

Answered: (a) Use the definition to find an…

AS Maths - Proof that integration relates to finding areas

When evaluating the definite integral of a vector function, what is actually kept constant? - Quora

Can the definite integral be used for purposes other than finding the area under a curve? If so, what are some examples and how do they work mathematically? - Quora

Calculus - Area under a Curve (video lessons, examples, solutions)

calculus - Why is the area under a curve the integral? - Mathematics Stack Exchange

Why can an integral be negative when it essentially symbolises the area under the curve of a function? - Quora