Is the area between the graph of $f\left(x\right)=\frac{1}{x}$ and the x-axis over the interval $\left[1,\text{+}\infty \right)$ finite or infinite? If this same region is revolved about the $x$-axis, is the volume finite or infinite? Surprisingly, the area of the region described is infinite, but the volume of the solid obtained by revolving this region about the $x$-axis is finite.

In this section, we define integrals over an infinite interval as well as integrals of functions containing a discontinuity on the interval. Integrals of these types are called improper integrals. We examine several techniques for evaluating improper integrals, all of which involve taking limits.