The Definition of the Definite Integral

Let f be defined on [a, b].

Partition [a, b] by choosing n + 1 points a = x₀ < x₁ < x₂ < ... < x_n-1 < x_n = b.

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

Let |P| 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.