Advanced Math Seminar

Improper Integrals

Improper Integrals

Improper integrals come in two varieties. Our main concern with these integrals is whether or not they converge to a finite number.

1. Infinite limits

When one or both of the limits of an integral are

we can rewrite the integral as a proper integral and combine it with a limit at infinity.

Example: Evaluate, if possible:

First we rewrite the integral as

Then we try to find an antiderivative and use the Fundamental Theorem.

Now we need to evaluate the limit. Since

as we get

2. Infinite integrand

Sometimes an improper integral can be easy to miss because you don't notice it's improper. The following integral looks fine until you realize that the integrand becomes infinite at one of the limits.

Example: Evaluate, if possible:

We can rewrite this using a one-sided limit then evaluate the limit.

Now evaluate the limit.

STEP 2002, Math III, #1: Find the area of the region between the curve

and the

-axis , for . What happens to this area as tends to infinity?

Find the volume of the solid obtained when the region between the curve

and the

-axis , for , is rotated through radians about the -axis. What happens to this volume as tends to infinity?

STEP 1998, Math I, #2: Show, by means of a suitable change of variable, or otherwise, that

Hence, or otherwise, show that

STEP 2012, Math II, #3: Show that, for any function (for which the integrals exist),

Hence evaluate

and, using the substitution

,

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