Improper Riemann Integrals

One very special type of Riemann integrals are called improper Riemann integrals. We define this type of integral below.

Definition: Let $f$ be a function defined on an unbounded interval $I = [a, \infty)$ where $a \in \mathbb{R}$ and assume that $f$ is Riemann integrable on $[a, b]$ for all $b > a$. Then the Improper Riemann Integral of $f$ on $I$ is denoted $\displaystyle{\int_a^{\infty} f(x) \: dx}$, and $f$ is said to be Improper Riemann Integrable on $I$ if $\displaystyle{\lim_{b \to \infty} \int_a^b f(x) \: dx}$ exists, in which case we denote $\displaystyle{\int_a^{\infty} f(x) \: dx = \lim_{b \to \infty} \int_a^b f(x) \: dx}$.

If instead $I = (-\infty, a]$ where $a \in \mathbb{R}$ then we define the corresponding improper Riemann integral analogously, i.e., $\displaystyle{\int_{-\infty}^a f(x) \: dx = \lim_{b \to -\infty} \int_{b}^a f(x) \: dx}$.

Furthermore, if $f$ is improper Riemann integrable on $(-\infty, a]$ and on $[a, \infty)$ then we say that $f$ is improper Riemann integrable on all of $\mathbb{R}$ and:

(1)

\begin{align} \quad \int_{-\infty}^{\infty} f(x) \: dx = \lim_{b \to -\infty} \int_{b}^a f(x) \: dx + \lim_{b \to \infty} \int_a^{b} f(x) \: dx \end{align}

For example, consider the function $f(x) = \frac{1}{x^2}$. Since $f$ is a nonnegative function on $[1, \infty)$, we can interpret the improper integral of $f$ on $[1, \infty)$ as the are trapped under $f$ and above the $x$-axis on $[1, \infty)$. This is graphed below:

We claim that $f$ is improper Riemann integrable on $[1, \infty)$. We have that:

(2)

\begin{align} \quad \lim_{b \to \infty} \int_1^b \frac{1}{x^2} \: dx &= \lim_{b \to \infty} \left [ -\frac{1}{x}\right ]_{1}^{b} \\ \quad &= \lim_{b \to \infty} \left [ -\frac{1}{b} + 1 \right ] \\ \quad &= 1 \end{align}

So $f$ is indeed improper Riemann integrable on $[1, \infty)$ and moreover:

(3)

\begin{align} \quad \int_1^{\infty} \frac{1}{x^2} \: dx = 1 \end{align}