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Lecture 14

# MTH 162 Lecture 14: 6.6 Improper Integrals Notes Premium

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School
Department
Mathematics
Course
MTH 162
Professor
Pachero
Semester
Fall

Description
MTH 162 Calculus II 6.6 Improper Integrals Notes L. Sterling Abstract Provide a generalization to each of the key terms listed in this section. Improper Integrals What are improper integrals? ▯ Improper integrals are integrals with either one or both conditions that aren’t actually met. ▯ Improper integrals are also deﬁned when it comes when limits either do or do not exist. – When a limit does exist, then that improper integral is convergent. – When a limit doesn’t exist, then that improper integral is divergent. Improper Integrals Deﬁnitions When t ▯ a R If you have tf (x)dx actually exist for every number that occurs when t ▯ a, then the following a would be the provided limit that will exist as a ﬁnite number: Z 1 Z t f (x)dx =t!1m f (x)dx a a When t ▯ b If you have tf (x)dx actually exist for every number that occurs when t ▯ b, then the following a would be the provided limit that will exist as a ﬁnite number: Z Z b b f (x)dx = lim f (x)dx ▯1 t!▯1 t R R It would considered convergent if the improper integrals)dx and b f (x)dx exist thanks a ▯1 to their corresponding limits; it will still be considered divergent if the limits don’t exist. From ▯1 to 1 R1 If you have▯1 f (x)dx is the improper integral that is being deﬁned as the following with a being any real number, then it would be considered to be convergent if both terms actually converge or divergent if both terms actually diverge: Z1 Z a Z 1 f (x)dx = f (x)dx + f (x)dx ▯1 ▯1 a Convergent and Divergent When you have the following, it can be convergent if p > 1 while divergent if p ▯ 1:
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