Definite Integrals

# Definite Integrals

We are now going to formalize a definition for the definite integral.

Definition: If $f$ is an integrable function over the closed interval $[a, b]$, then $\int_a^b f(x) \: dx = \lim_{n \to \infty} \sum_{i = 1}^n f(x_i) \Delta x$ where $\Delta x = \frac{b - a}{n}$ is the width of the subintervals of $[a, b]$, $x_i = a + i\Delta x = a + i\frac{b - a}{n}$ is the x-coordinate of any $i^\mathrm{th}$ subinterval, and $f(x_i)$ is the height of the $i^\mathrm{th}$ subinterval. For every $\epsilon > 0$ there exists an integer $N$ such that $\biggr \rvert \int_a^b f(x) \: dx - \sum_{i=1}^n f(x_i)\Delta x \biggr \rvert < \epsilon$ for every integer $n > N$ and for every $x_i \in [x_{i - 1}, x_i + 1]$. |