The basis of the
Riemann integral. Given a function f:[a,b]->
R, a
partition P={a=x
0<x
1<...<x
n=b}
of the
interval [a,b], and points C={c
i∈[x
i-1,x
i] : i=1,...,n}, we wish to approximate (what will eventually become) the Riemann integral ∫
ab f(x)dx.
The Riemann sum of f with respect to the above partition {xi} and points {ci} is the sum
I(f;x0,...,xn;c1,...,cn) =
∑i=1n (xi-xi-1)f(ci)
That is,
approximate the value of f in each subinterval of the partition by its value at some point in the subinterval
(this might make sense e.g. if f is continuous), and replace the area under f in the subinterval with the area of the
rectangle of that height on the interval.
Define d(P)=max {xi-xi-1: i=1,...,n}.
The Riemann integral exists iff there exists a limit for the values I(f;P;C) as d(P) -> 0. That is, there is some value I such that for any ε>0 there exists some δ>0 such that if d(P)<δ and we pick any points ci ∈ [xi-1,xi] we have that
|I-I(f;P;C)| < ε.
In such a case, I is the value of the Riemann integral.
In other words, we can guarantee that a Riemann sum will be a good approximation of the integral of a (Riemann-integrable!) function merely by forcing the partition to be fine enough.