Understanding integration

While I do understand that

lim(n->(infinity) epsilon(i=1,n)f(xi)*(delta)x

should give an area under f(x), I fail to understand why this equation equals F(b)-F(a). I mean, why is the antiderivate interpreted as the area under a curve? I've looked in my math-book and searched wikipedia, but they only write that the closed integral is defined as the above formula, with no further explanation. This has been driving me crazy lately, if anyone could explain or post a link to an explanation I would greatly appreciate it.