Introduction to Improper Integrals

Is the area between the graph of [latex]f\left(x\right)=\frac{1}{x}[/latex] and the x-axis over the interval [latex]\left[1,\text{+}\infty \right)[/latex] finite or infinite? If this same region is revolved about the [latex]x[/latex]-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 [latex]x[/latex]-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.