LectureNotesMATH150BSec6_6.pdf - Sec 6.6 Improper Integrals MATH 150B Lecture Notes Topics Covered Page Introduction 1 Improper integrals of Type 1 1

LectureNotesMATH150BSec6_6.pdf - Sec 6.6 Improper Integrals...

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Sec 6.6: Improper Integrals MATH 150B: Lecture Notes Topics Covered Page Introduction 1 Improper integrals of Type 1 1 The p - test 3 Improper integrals of Type 2: Discontinuous integrands 4 Comparaison Test 5 Introduction In the previous sections, we have covered definite integrals. That is, integrals of types Z b a f ( x ) dx where the interval [ a, b ] is a finite domaine and that f ( x ) is continuous on [ a, b ]. In this section, we extend this concept to investigate integrals where the function f ( x ) may have an infinite discontinuity (vertical asymptotes) and on infinite intervals. Examples of such integrals are as follows: a ) Z 1 0 1 x 4 dx, b ) Z 1 e - x 2 dx, c ) Z -∞ 1 1 + x 4 dx. + Such integrals are called improper integrals . x y O a x y O a Definition 1. Improper integrals are said to be: convergent if the limit is finite and that limit is the value of the improper integral. divergent if the limit does not exist. Improper integrals of Type 1 + If Z t a f ( x ) dx exists for every number t a , then Z a f ( x ) dx = lim t →∞ Z t a f ( x ) dx, c D. Bichara

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