Integrals

Overview

Let $s$ be a step function defined on interval $[a,b]$, and let $P=\{x_0,x_1,\dots ,x_n\}$ be a partition of $[a,b]$ such that $s$ is constant on the open subintervals of $P$. Denote by $s_k$ the constant value that $s$ takes in the $k$th open subinterval, so that $s(x)=s_k\text{if}{x}_{k-1}<x<x_k,k=1,2,\dots ,n.$

The integral of $s$ from $a$ to $b$, denoted by the symbol ${\int }_a^{b}s(x)dx$, is defined by the following formula: ${\int }_a^{b}s(x)dx={\sum }_{k=1}^n{s}_k\cdot (x_k-x_{k-1})$

Bibliography

• Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).