8.4 day one. Improper Integrals. Greg Kelly, Hanford High School, Richland, Washington. Sometimes we can find integrals for functions where the function or the limits are infinite. These are called improper integrals .
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
8.4 day one
Greg Kelly, Hanford High School, Richland, Washington
Sometimes we can find integrals for functions where the function or the limits are infinite. These are called improper integrals.
Until now we have been finding integrals of continuous functions over closed intervals.
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.
(right hand limit)
We approach the limit from inside the interval.
This integral diverges.
The function approaches
If then gets bigger and bigger as , therefore the integral diverges.
If then b has a negative exponent and ,
therefore the integral converges.
(P is a constant.)
What happens here?