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Why is the area under a curve the integral?

Why does calculating an integral or an anti-derivatives represent an area? How do they figure it out? What is the relationship between them?

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marked as duplicate by J. M., Pedro Tamaroff, Jonas Teuwen, MJD, Zev Chonoles Jul 14 '12 at 2:33

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A textbook on calculus will have some explanation and intuitive motivation for this. –  GEdgar Jul 13 '12 at 23:30

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This is basic level stuff covered in first course of calculus. You may also refer to this:

http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Geometric_intuition

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