Definition:Improper Integral/Unbounded Closed Interval

Definition

Unbounded Above

Let $f$ be a real function which is continuous on the unbounded closed interval $\hointr a \to$.

Then the improper integral of $f$ over $\hointr a \to$ is defined as:

$\ds \int_a^{\mathop \to +\infty} \map f t \rd t := \lim_{\gamma \mathop \to +\infty} \int_a^\gamma \map f t \rd t$

Unbounded Below

Let $f$ be a real function which is continuous on the unbounded closed interval $\hointl \gets b$.

Then the improper integral of $f$ over $\hointl \gets b$ is defined as:

$\ds \int_{\mathop \to -\infty}^b \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^b \map f t \rd t$