Improper Integrals 8.3 day one Greg Kelly, Hanford High School, Richland, Washington

Until now we have been finding integrals of continuous functions over closed intervals. Sometimes we can find integrals for functions where the function or the limits are infinite. These are called improper integrals.

Since the limit is undefined or infinity, we say that this integral diverges. Is it possible that an improper integral does not diverge?

Let’s try a different integral. Since the limit is 1, then the area approaches 1 as x goes to infinity. We say that this integral converges.

Example 1: The function is undefined at x = 1 . Can we find the area under an infinitely high curve? Since x = 1 is an asymptote, the function has no maximum. We could define this integral as: (left hand limit) We must approach the limit from inside the interval.

Rationalize the numerator.

This integral converges because it approaches a solution.

Example 2: (right hand limit) We approach the limit from inside the interval. This integral diverges.

Example 3: The function approaches when .

p Example 4: (P is a constant.) What happens here? If then gets bigger and bigger as , therefore the integral diverges. If then b has a negative exponent and , therefore the integral converges. p