Carrotsticks
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The reason why the integral (sometimes called the 'anti-derivative') gives the area under the curve, is from the Fundamental Theorem of Calculus and something called 'Riemann Sums' (that is what deswa1 and D94 were talking about with the rectangles).
http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
However, the proof for this is well beyond the scope of even the Extension 2 course and it is one of the few things in HSC Mathematics that you will have to take for granted.
http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
However, the proof for this is well beyond the scope of even the Extension 2 course and it is one of the few things in HSC Mathematics that you will have to take for granted.