1.41 Improper integrals

Improper integrals. An integral $\int_{a}^{b}f(x)\,dx$ is called improper if either $f$ is unbounded or $(a,b)$ is an infinite interval.

Suppose that $f$ is a continuous on $[a,\infty).$ Then for each $R$ we can form $\int_{a}^{R}f(x)\,dx.$ We define the improper integral of $f$ over $[a,\infty)$ to be

where this limit exists; otherwise, we say that the integral $\int_{a}^{\infty}f(x)\,dx$ diverges. Recall that limits are real numbers, so that ‘converges’ means ‘tends to a real number’. One way in which an integral can diverge is for there to be an infinite area under the graph. When the integral converges, it represents the area under the graph of $f$ over the range $a\leq x<\infty$. In calculations, we start by considering $\int_{a}^{R}f(x)\,dx$ and later consider the limits as $R\rightarrow\infty$.