Integration and area under curve

[jmsdg7]

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 don't understand how that can equal the area under a curve.

this whole question might be pointless because we haven't learned how to evaluate definite integrals yet and maybe my answer lies in that.

Any 2 cents worth is appreciated!

 

[l'Hôpital]

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