Integration and area under curve

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I have seen a thread with a similair title but passed up on what i want to know.

I just want somebody to explain to me why definite integration equals the area under between the function and the x axis

Ive just been through indefinite integration, then using the summation formulas in my AP calculus class, and i understand the concept of going from acceleration to velocity to position, but i dont understand how that can equal the area under a curve.

this whole question might be pointless because we havent learned how to evaluate definite integrals yet and maybe my answer lies in that.

Any 2 cents worth is appreciated!!!
 

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