Riemann Sum (thing)

The basis of the Riemann integral. Given a function f:[a,b]->R, a partition P={a=x₀<x₁<...<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 ∫_a^b f(x)dx.

The Riemann sum of f with respect to the above partition {x_i} and points {c_i} is the sum

I(f;x₀,...,x_n;c₁,...,c_n) = ∑_i=1ⁿ (x_i-x_i-1)f(c_i)

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 {x_i-x_i-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 c_i ∈ [x_i-1,x_i] 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.