7.8 Improper Integrals

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 2: (right hand limit) We approach the limit from inside the interval.

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

Example 3: The function approaches when .

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.

p. 547 #’s 1-7 odd, 9-41 every other odd

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