8.4 day one: Improper Integrals Durango & Silverton Railroad, Durango, Colorado Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2010

8.4 day one: Improper Integrals Durango & Silverton Railroad, Durango, Colorado Greg Kelly, Hanford High School, Richland, Washington Photo by Greg Kelly, 2010

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.

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