Calculus 1000A/B Lecture 6: improper

• School: Western University
• Department: Calculus
• Course: Calculus 1000A/B
• Professor: Alan Mac Isaac

Document Summary

So far we dealt with integration of continuous functions on bounded intervals. In this section we will discuss integration of continuous functions on unbounded intervals, and also integration of certain unbounded functions. Suppose a function f (x) is continuous on the interval (a, ), so that the integral r b a f (x)dx is well-de ned for any b > a. The limit of this integral as b will be called the improper integral of f (x) on (a, ). That is b z b (3. 1) f (x)dx = lim. If the limit exists (i. e. , it is a nite number), then we say that the integral r a f (x)dx converges or is convergent. If the limit is in nite, or does not exist, we say that the improper integral diverges or is divergent. 1 + x2 = tan 1 x(cid:12)(cid:12)(cid:12) b z b. 1 + x2 = lim b tan 1 b =