如何證明 integration (積分) 可用來計算「面積」？

In fact, in general, the value of definite integral in most calculation does not represesnt the area of a region. Instead, the definite integral is actually the limit of a sum.

I don't intend to give any rigorous proof for integral here (since you will learn that in F.6 pure maths or university level maths). However, i will sketch the reason here.

Consider a non-negative function y = f(x) on [a, b], the limit of sum is the 'area' bounded by f(x), the axis x = a and x = b, and x-axis.
So lim Σf(xi)(xi - xi+1) (i from 1 to n) will leads to actual area
[here, xi are the point lying between a and b, we select n tends to infinity, consequently the division of the interval by x1, x2, ... will be much smaller and so the result is obtained.]

這是一個很偉大的Theorem Result

它的名字是fundamental theorem of calculus.

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