The Definition of the Definite Integral

 

Let f be defined on [a, b].

Partition [a, b] into n not-necessarily equal subintervals by choosing n + 1 points a = x0 < x1 < x2 < ... < xn-1 < xn = b. 

Let Δx1, Δx2, ..., Δxn denote the lengths of the resulting subintervals.

Let maxΔxk denote the norm of the partition. 

Choose n arbitrary points , one in each subinterval. 

The definite integral of f from a to b, denoted by , is defined to be

provided this limit exists.