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# Class Note for MATH 1432 at UH 3

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mm mm the memoquot pm m mama maqueg um mm Swank a Mama unmons 7 m mm M mmquot M u lt lt b 5 mm a WWW hm mm 39x 5 mm W M By W may mm mm the memoquot pm m mama maqueg um w 131A 72quot mm mmquot 2 3 is x u 39x g a w Ag 01 7 m mm M mmquot M u lt lt b 5 mm a WWW hm mm 39x 5 mm W M By W may mm mm the memoquot pm m mama maqueg um w 131A 72quot mm mmquot 2 3 is x u 39x g a w A gdxwlgdx a 7 m mm M mmquot M u lt lt b 5 mm a WWW hm mm 39x 5 mm W M By W may mm mm the memoquot pm m mama maqueg um w 131A 72quot mm mmquot 2 3 is x u 39x g a w 1 1 1 AWWJM SWAN 7 m mm M mmquot M u lt lt b 5 mm a WWW hm mm 39x 5 mm W M By W may mm mm the memoquot pm m mama maqueg um w 131A 72quot mm mmquot 2 3 is x u 39x g a w 1 1 1 AWWJM SWAN Unbounded Intervals Unbounded Functions Integrals Over Unbounded Intervals ylk Improper Integrals 39 Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 1 7 Unbounded Intervals Unbounded Functions Integrals Over Unbounded Intervals ylk Improper Integrals Let f be continuous on 3 oo Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 17 Unbounded Intervals Unbounded Functions Integrals Over Unbounded Intervals Improper Integrals ylk Let f be continuous on 3 oo We define M fX dX gt a a b x Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 17 Unbounded Intervals Unbounded Functions Integrals Over Unbounded Interval ylk Improper Integrals Let f be continuous on 3 oo We define 0 rx dX bigtmoo b rx dX Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 17 Unbounded Intervals Unbounded Functions Integrals Over Unbounded Interval Improper Integrals y A Let f be continuous on 3 oo We define f 00 b fX dX lim fX dX gt a b gtoo a a b The improper integral converges if the limit exists I Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 17 Unbounded Intervals Unbounded Functions Integrals Over Unbounded Interval yll Improper Integrals Let f be continuous on 3 00 We define 0 rx dX DH moo b rx dX The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist a Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 1739 Unbounded Intervals Unbounded Functions Integrals Over Unbounded Intervals yll Improper Integrals Let f be continuous on 3 00 We define 0 rx dX DH moo b rx dX The improper integral converges if the limit exists a The improper integral diverges if the limit doesn t exist If 3me rx 7t 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 1739 Unbounded Intervals Unbounded Functions Integrals Over Unbounded Intervals yll Improper Integrals Let f be continuous on 3 00 We define 0 rx dX DH moo b rx dX The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist a If blim fX 72 0 then fX dX diverges H Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 1739 Unbounded Intervals Unbounded Functions Integrals Over Unbounded Intervals yll Improper Integrals Let f be continuous on 3 00 We define f 00 b M fX dX lim fX dX b c a b oo a a The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist If blim fX 72 0 then fX dX diverges H gtOO a If f is continuous on 00 b Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 1739 Unbounded Intervals Unbounded Functions Integrals Over Unbounded Intervals yll Improper Integrals Let f be continuous on 3 00 We define f 00 b M fX dX lim fX dX b c a b oo a a The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist If blim fX 72 0 then fX dX diverges H gtOO a b If f is continuous on 00 b fX dX Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 1739 n Unbounded Intervals Unbounded Functions Integrals Over Unbounded Intervals yll Improper Integrals Let f be continuous on 3 00 We define f 00 b M fX dX lim fX dX b c a b oo a a The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist If blim fX 72 0 then fX dX diverges l b b If f is continuous on 00 b fX dX lim fX dX OO a gt oo a Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 I7 n Unbounded Intervals Unbounded Functions Integrals Over Unbounded Intervals yll Improper Integrals Let f be continuous on 3 00 We define f 00 b M fX dX lim fX dX b c a b oo a a The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist If blim fX 72 0 then fX dX diverges l b b If f is continuous on 00 b fX dX lim fX dX OO a gt oo a If f is cont on oooo H Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 I7 n Unbounded Intervals Unbounded Functions Integrals Over Unbounded Intervals yll Improper Integrals Let f be continuous on 3 00 We define f 00 b M fX dX lim fX dX b c a b oo a a The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist If blim fX 72 0 then fX dX diverges l b b If f is continuous on 00 b fX dX lim fX dX OO a gt oo a If f is cont on oooo fX dX H Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 I7 Improper Integrals Unbounded Intervals Unbounded Functions Integrals Over Unbounded Intervals J A Let f be continuous on 3 00 We define f 00 b M fX dX lim fX dX b a b oo a a The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist If blim fX 72 0 then fX dX diverges l b b If f is continuous on 00 b fX dX lim fX dX OO a gt oo a If f is cont on oooo dX X dX I OOltX dX H Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 3 I7 m x mm cum a l ix A 5i n m is 3L x 3 E ix 5 Hi 33 a n N 2 an n g j A will A Q gags z a n N 2 an n g j Ai rw n 9 anime xi 5 n N 2 an n g a x A 3 33 as x 3 3 5 8533 Awi nn j 3 33 as x 3 3 5 8533 3 ii A w is 3L n 3 E 3 33 as x 3 3 5 8533 cf 5 xii gunisd ii A w is 3L n 3 E 3 33 as x 3 3 5 8533 x 5 3 n 3 E cf 5 xii gunisd ii A w is 3L n 3 E 3 33 as x 3 3 5 8533 XV 3 ii 5 31 n 3 E cf 5 xii gunisd ii A w is 3L n 3 E 3 33 as x 3 3 5 8533 XV 3 3Q n XV 3N a 5 31 n 3 E cf 5 xii gunisd ii A w is 3L n 3 E 3 33 as x 3 3 5 8533 Q5 saga Q9 if 55 n ix txxv nism a 5 31 n 3 E cf 5 xii gunisd ii A w is 3L n 3 E 3 33 as x 3 3 5 8533 3 g A 1 3531 5 3 g x E 327 2 n A n 3Q E A via 3 9 i 33 7 5i n m is 3L n 3 E gamma 130an gamma 130an g W 130an m Who 5 mm m the man WW Who 5 mm m the man WW w 2x AW wm sWWWmmmnwww f 2 4km AL mug w 41de Who 5 mm m the man WW 3 w mm w 2x A 2x waz leZu Mm Who 5 mm m the man WW 3 w mm W m 7m Wm w 2x A 2x inwm hm w 2x 2x A 7 mom M W m 7m Wm m 41934 hm w 2x 2x A 7 mom M W m 7m Wm m 41934 hm w a Nmenutwed dnmda me rm hm a w 2x 2x A 7 mom M W m 7m Wm m 41934 hm w a Nmenutwed dnmda me rm hm a w mm L 2x 2x A 7 mom M W m 7m Wm m 41934 hm w a W W m aw is a f 4x 4x mm Su pm 2x 2x A 7 mom M W m 7m Wm m 41934 hm w a W W m aw is a f 4x 4x mm Su pm WWW 2x 2x A 7 mom M W m 7m Wm m 41934 hm w a W W m aw is a f 4x 4x mm Su pm WWW L 2x 2x A 7 mom M W m 7m Wm m 41934 m w a W W m aw is a f 4x 4x mm Su pm Mpw f m m A m m 43114qu u 2x 2x A 7 mom M W m 7m Wm m 41934 m w a W W m aw is a f 4x 4x mm Su pm Mpw f m m A m m 43114qu u mm mm mmmsw mm 31 Who 5 mm m the man WW Mmuju W W m aw is pkg1m 4x Wm sixMM amp39xxL pm a m m A m m W AM a z u maths mm mmmsecm x 41x 91 Who 5 mm m the man WW 2x 5 2x W lzuJW w 2x n u a Nmeuutwed dnmda me u hm 39Wx w 4 mm SanWm amp39xxL 3W L m m A m m 43114qu a mm mm mmmsecmme ll Who 5 mm m the man WW 2x 5 2x W lzuJW w 2x w n Wampn1 n a W W m aw is pkg1m 4x mm SanWm amp39xxL 3W L m m A m m 43114qu a mm mm mmmsecmme ll Who 5 mm m the man WW 2x 5 2x W lzuJW Aw iixxza mum zgwnumwn a W W m aw is pkg1m 4x mm SanWm amp39xxL 3W L m m A m m 43114qu a mm mm mmmsecmme ll Who 5 mm m the man WW 2x 5 2x W lzuJW Aw iixxza mum J un1rn1w a W W m aw is pkg1m 4x mm SanWm amp39xxL 3W L m m A m m 43114qu a mm mm mmmsecmme ll WM 5WW mammngmmw 1 2x 2x deiamp4W D 2 dxzwnunz 27 humpquot sz a W W m aw is pkg1m 4x mm SanWm amp39xxL 3W L m m A m m 43114qu a mm mm mmmsecmme ll Improper Integrals idxozgt0 X06 J A xv Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 7 17 Improper Integrals idxozgt0 X06 J A 1XO xv Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 7 17 Improper Integrals idxozgt0 X06 J A 1XO xv Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 7 17 Improper Integrals idxozgt0 X06 J A ifozgt1 00 ifagl xv Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 7 17 Im proper Integrals idxozgt0 X06 J A 1XO xv Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 7 17 Im proper Integrals idxozgt0 X06 J A Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 7 17 Im proper Integrals idxozgt0 X06 J A OO7 ifozgt1 ifagl Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 717 Im proper Integrals idxozgt0 quot ylk 1 I oo 1 fOzgt1 dX O 1 1 XO oo ifagl 1O OO 1 Iim X1O 1 1 1 OzX gtoo 1 04 ifagt1 oo ifozlt1 Math 1432 Section 26626 Lecture 20 March 27 2008 7 17 Jiwen He University of Houston Improper Integrals w 39 39 quot ia39XozgtO X06 J A If 04 1 then Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 7 17 Improper Integrals s39 39 dXozgtO Jiwen He University of Houston 39 Math 1432 Section 26626 Lecture 20 March 27 2008 7 17 Improper Integrals s39 39 dXozgtO Ifozzl then 01 dX nXOO i X 1 i Jiwen He University of Houston 39 Math 1432 Section 26626 Lecture 20 March 27 2008 7 17 Improper Integrals s39 39 dXozgtO 01 f1 dXnXlt1O XigtmOOInX n1 H Jiwen He University of Houston 39 Math 1432 Section 26626 Lecture 20 March 27 2008 7 17 Improper Integrals s39 39 dXozgtO Ifozzl then 01 dX nXlt1gtO Iim InX n1oo 1 X X gtOO Jiwen He University of Houston 39 Math 1432 Section 26626 Lecture 20 March 27 2008 7 17 mg 2W 5 GM 2 rm 1 2W 5 GM Improper Integrals Unbounded Intervals Unbounded Functions Inegrals of Unounded Functions VA Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 9 l7 m proper Integrals i a 124 n 7 U n bou nded Intervals U n bou nded Fu nctions Inegrals of Unbounded Functions Let f be continuous on 3 b Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 9 17 m proper Integrals i a 124 n 7 U n bou nded Intervals U n bou nded Fu nctions Inegrals of Unbounded Functions Let f be continuous on 3 b but fX a ioo 39 as X a b x219 Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 9 17 m proper Integrals i a 124 n 7 U n bou nded Intervals U n bou nded Fu nctions Inegrals of Unbounded Functions Let f be continuous on 3 b but fX a ioo 39 as X a b We define f b fX dX x219 Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 9 17 m proper Integrals i a ism n 7 U n bou nded Intervals U n bou nded Fu nctions Inegrals of Unbounded Functions Let f be continuous on 3 b but fX a ioo 39 j as X a b We define f b C fX dX Iim fX dx 6 39x 7 a C gtb a j The improper integral converges if the limit exists Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 9 17 m proper lntegra ls p n U n bou nded Intervals U n bou nded Fu nctions Integrals of Unbounded Functions Let f be continuous on a b but fX a ioo 39 j as X a b We define f b C fX dX lirB fX dX The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist If f is continuous on a b but fX a ioo as X a a Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 9 17 m proper lntegra ls p n U n bou nded Intervals U n bou nded Fu nctions Integrals of Unbounded Functions Let f be continuous on a b but fX a ioo 39 j as X a b We define f b C fX dX lirB fX dX The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist If f is continuous on a b but fX a ioo as X a a fab ax dx Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 9 17 m proper lntegra ls p n U n bou nded Intervals U n bou nded Fu nctions Integrals of Unbounded Functions Let f be continuous on a b but fX a ioo 39 j as X a b We define f b C fX dX lirB fX dX The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist If f is continuous on a b but fX a ioo as X a a fab ax dx Cir b ax dx Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 9 17 m proper lntegra ls p Iiiiii U n bou nded Intervals U n bou nded Fu nctions Integrals of Unbounded Functions Let f be continuous on a b but fX a ioo j as X a b We define f b C fXdX lirB fXdX The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist If f is continuous on a b but fX a ioo as X a a fab ax dx Cir b ax dx If f is cont on a b Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 m proper lntegra ls i U n bou nded Intervals U n bou nded Fu nctions Integrals of Unbounded Functions Let f be continuous on a b but fx a ioo j as X a b We define f b C fxdx lirB fxdx The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist If f is continuous on a b but fx a ioo as X a a fab ax dx Cir b fx dx If f is cont on a b except at some point c in a b where fx ioo asx c orchJF fab fx dx llli Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 m proper lntegra ls i U n bou nded Intervals U n bou nded Fu nctions Integrals of Unbounded Functions Let f be continuous on a b but fx a ioo j as X a b We define f b C fxdx lirB fxdx The improper integral converges if the limit exists The improper integral diverges if the limit doesn t exist If f is continuous on a b but fx a ioo as X a a fab ax dx Cir b fx dx If f is cont on a b except at some point c in a b where fx ioo asx c orchJF abfxdxlcfxdxbfxdx llli Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 z r x w i r A E 2 w 5 3 n A E 6 z r x w i r A E 2 w 5 3 n A E a 333 n XV 3 a z r x w i r A E 2 w 5 3 n A E 3327 333 n XV 3 a z r x w i r A E 2 w 5 3 n A E i n 3 3 a 3 x r 5 n 9 n XV Q 2 r x w i r 3 E 2 w is 3 n 3 E m5 32 as ea n 3 3 a 3 x 3 5 n 9 n XV 3 z r x w i r 3 E 2 w is 3 n 3 E m r x w i r A 25 3 3n n A E m5 32 as ea n 3 3 a z r x w i r A E 2 w 5 3 n A E 3 x 3 5 n 9 n XV 3 XV Q N m r551 r 3 E rigs 3a A E a m5 32 as i n 3 3 31 5 Marn k a z r551 r 3 2 Asia 3 A E 325 2 n a n x 3 a m r x w i r A 25 3 3n n A E m5 32 as i n 3 3 31 5 Marn k a z r551 r 3 2 Asia 3 A E 1 Am E E if 7 nx k a N a W Ame 5 WW N 2x a Km E E if 7 nx k a N a W Ame 5 WW N 2x a Km 313 Enfnx k a N a g l N 2x a Km XEEM 313 Enfnx k a N a g l N 2x a Km mi 12 313 Enfnx k a N a g l N 2x a Km iii mixinth E E if 7 nx k BIN N a W Ame E W 21 xii 5223 mixinth E E if 7 nx k BIN N a W Ame E W 21 xii in zHT muzm memzd xm 2541 I muzm memzd xm 2541 I XV k aRE Wu g Txiime s a f E gurg ix 2 ix g k Improper Integrals ii Ii i What is Wrong with This 1 I l fm x 22 Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 13 17 Improper Integrals i r qu What is Wrong with This What is wrong with the following argument H x Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 1317 wI39I Improper Integrals ii I What is Wrong with This 1 II 22 I I fx I I Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 13 17 wI39I Improper Integrals ii I What is Wrong with This 1 II 22 I I fx I I Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 13 17 wI39T Improper Integrals ii I What is Wrong with This 4 1 1 4 3 2dx 1X 2 X 21 2 1 1 22 I I fx I I Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 13 17 a n Improper Integrals ii r What is Wrong with This 4 1 1 4 3 2dx 1X 2 X 21 2 Note that w a oo asx 2 or X x 22 X 2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 Ii What is Wrong with This Improper Integrals i r l NotethataooaSXa2 or X l fx 4 1 I H X a 2 To evaluate 2 dX we 1 X 2 1 need to calculate the improper integrals fill 21 4 1 dear dzmdx39 Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 2739 2008 13 17 Improper Integrals i r Ii What is Wrong with This What is wrong with the following argument 4 4 1 1 3 f zdx 1X 2 X 21 2 i NotethataooaSXa2 or X l fx 4 I H X a 2 To evaluate 2 dX we 1 X 2 A need to calculate the improper integrals 1 x 2 1 4 i 392 i dear dzmdx39 gtgr Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 2739 2008 13 17 Improper Integrals i r Ii What is Wrong with This What is wrong with the following argument 4 4 1 1 3 d 1 X 22 X 7g X 2 1 2 2 1 d 1ltx 2gt2 X Note that w a oo asx 2 or x 4 4 H X a 2 To evaluate 2 dX we 1 X 2 A need to calculate the improper integrals l I I 2 1 d d 4 1 d 2 3 4 x 2gt2 X3 x 2gt2 X39 Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 13 17 Improper Integrals i r Ii What is Wrong with This What is wrong with the following argument 4 4 1 1 3 dxg 1 X 22 X 2 1 2 2 2 1 1 d 1 X 22 X X 2 1 Note that w a oo asx 2 or x 4 4 0622 X a 2 To evaluate 2 dX we 1 X 2 A need to calculate the improper integrals l I I 2 1 d d 4 1 d 2 3 4 x 2gt2 X3 x 2gt2 X39 Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 13 17 Improper Integrals i r Ii What is Wrong with This What is wrong with the following argument 4 1 1 4 3 d 1X 22 X7 X 2 Note that w a oo asx 2 or x 4 4 H X a 2 To evaluate 2 dX we 1 X 2 A need to calculate the improper integrals l I I 2 1 d d 4 1 d 2 3 4 x 2gt2 X3 x 2gt2 X39 Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 13 17 Improper Integrals i r Ii What is Wrong with This What is wrong with the following argument 4 1 1 4 3 d 1X 22 X7 X 2 Note that w a oo asx 2 or x 4 4 H X a 2 To evaluate 2 dX we 1 X 2 A need to calculate the improper integrals l I I 2 1 d d 4 1 d 2 3 4 x 2gt2 X3 x 2gt2 X39 Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 13 17 Improper Integrals 39 quot Unbounded Intervals Unbounded Functions Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 14 17 Improper Integrals 39 quot Unbounded Intervals Unbounded Functions Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 14 17 Improper Integrals v r Unbounded Intervals Unbounded Functions Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 14 17 Improper Integrals or Unbounded Intervals 1 1 04 7 007 U nbounded Fu nctions ifalt1 ifozZl Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 14 17 Improper Integrals v r Unbounded Intervals Unbounded Functions 1 ifozlt1 7 7 dX 1O 0 oo ifozZl or If 04 75 1 then 1 1 dX 0 XO Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 14 17 Im proper Integrals Unbounded Intervals Unbounded Functions 7 CO7 ifalt1 ifozZl Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 14 17 Improper Integrals v r Unbounded Intervals Unbounded Functions ifalt1 ifozZl Jiwen He University of Houston 39 Math 1432 Section 26626 Lecture 20 March 27 2008 Improper Integrals w r Unbounded Intervals Unbounded Functions 1 1 1 7 dX 1O 0 XO 00 c0421 fa7 1then 1 1 1 1 1 1 dX X1O Iim X1O O X04 1 a O 1 a 1 O X gtO Jiwen He University of Houston 39 Math 1432 Section 26626 Lecture 20 March 27 2008 Improper Integrals w r Unbounded Intervals Unbounded Functions 1 1 1 7 dX 1O 0 XO oo Ifoz21 fa7 1then 1 1 1 1 1 dX X1O Iim X1O O X04 1 a O 1 a 1 O X gtO If 04 1 then Jiwen He University of Houston 1 Math 1432 Section 26626 Lecture 20 March 27 2008 14 17 Improper Integrals w r Unbounded Intervals Unbounded Functions 1 1 1 7 dX 1O 0 XO oo Ifoz21 fa7 1then 1 1 1 1 1 dX X1O Iim X1O O X04 1 a O 1 a 1 O X gtO If 04 1 then Jiwen He University of Houston 1 Math 1432 Section 26626 Lecture 20 March 27 2008 14 17 Improper Integrals w r Unbounded Intervals Unbounded Functions 1 1 1 7 dX 1O 0 XO oo Ifoz21 fa7 1then 1 1 1 1 1 dX X1O Iim X1O O X04 1 a O 1 a 1 O X gtO If 04 1 then 1 1 1 d 1 39m fox X nXO n XOnX H Jiwen He University of Houston 1 Math 1432 Section 26626 Lecture 20 March 27 2008 14 17 Improper Integrals w r Unbounded Intervals Unbounded Functions 1 1 1 7 dX 1O 0 XO oo Ifoz21 fa7 1then 1 1 1 1 1 dX X1O Iim X1O O X04 1 a O 1 a 1 O X gtO ifalt1 oo ifozgt1 foz1then 11 1 d I I1 I39 fox X nXO n XLFT8nX oo H Jiwen He University of Houston 1 Math 1432 Section 26626 Lecture 20 March 27 2008 14 17 3 1wquot 2W 5 GM 39i 39 Comparison Test Comparison Test for Conver g RV Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 16 17 Comparison Test Comparison Test for Conver y Let be improper integration over an I g I I interval I I Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 16 17 View Comparison Test Comparison Test for Conver y Let be improper integration over an I g I f interval I and suppose that f and g are V continuous on I I gt Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 16 17 I 1quot ii Comparison Test Comparison Test for Conver y Let be improper integration over an I g I f interval I and suppose that f and g are V continuous on I such that id 3 ogrxggx vxe Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 16 17 View Comparison Test Comparison Test for Conver y Let be improper integration over an I g I f interval I and suppose that f and g are V continuous on I such that id 3 ogrxggx vxe o If gx dX converges I Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 16 17 View Comparison Test Comparison Test for Conver y Let be improper integration over an I g I f interval I and suppose that f and g are V continuous on I such that id 3 ogrxggx vxe o If gx dX converges then so does fX dX I I Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 16 17 I I I Comparison Test Comparison Test for Conver y Let be improper integration over an I g I 39 f interval I and suppose that f and g are V continuous on I such that id 3 ogrxggx Vxel o If gx dX converges then so does fX dX I I o If fx dX diverges I Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 16 17 Comparison Test Comparison Test for Conver y Let be improper integration over an I g I 39 f interval I and suppose that f and g are V continuous on I such that Id 3 ogrxggx Vxel o If gx dX converges then so does fX dX I I o If fx dX diverges then so does gx dX I I Jiwen He University of Houston Math 1432 Section 26626 Lecture 20 March 27 2008 16 17 I lt1 I Comparison Test Comparison Test for Conver y Let be improper integration over an I g I 39 f interval I and suppose that f and g are V continuous on I such that Ia 3 ogrxggx Vxel o If gx dX converges then so does fX dX I I o If fx dX diverges then so does gx dX I I OO 1 The im ro er inte ra dX conver es since P P g 1 1X3 g Jiwen He University of Houston Math 1432 F Section 26626 Lecture 20 March 27 2008 I lt1 I Comparison Test Comparison Test for Conver y Let be improper integration over an I g I 39 f interval I and suppose that f and g are V continuous on I such that Ia 3 ogrxggx Vxel o If gx dX converges then so does fX dX I I o If fx dX diverges then so does gx dX I I OO 1 The im ro er inte ra dX conver es since P P g 1 1X3 g 1 1 Olt lt szl x 1X3 X37 Jiwen He University of Houston Math 1432 F Section 26626 Lecture 20 March 27 2008 I lt1 I Comparison Test Comparison Test for Conver y Let be improper integration over an I g I 39 f interval I and suppose that f and g are V continuous on I such that Ia 3 ogrxggx Vxel o If gx dX converges then so does fX dX I I o If fx dX diverges then so does gx dX I I OO 1 The im ro er inte ra dX conver es since P P g 1 1X3 g 1 1 00 1 O lt lt 3 VX Z 1 and dX converges 1 1X3 X M Jiwen He University of Houston Math 1432 F Section 26626 Lecture 20 March 27 2008 I lt1 I Comparison Test Comparison Test for Conver y Let be improper integration over an I g I 39 f interval I and suppose that f and g are V continuous on I such that Ia 3 ogrxggx Vxel o If gx dX converges then so does fX dX I I o If fx dX diverges then so does gx dX I I OO 1 The im ro er inte ra dX diver es since P P g 1 1X2 g Jiwen He University of Houston Math 1432 F Section 26626 Lecture 20 March 27 2008 I lt1 I Comparison Test Comparison Test for Conver y Let be improper integration over an I g I 39 f interval I and suppose that f and g are V continuous on I such that Ia 3 ogrxggx Vxel o If gx dX converges then so does fX dX I I o If fx dX diverges then so does gx dX I I OO 1 The im ro er inte ra dX diver es since P P g 1 1X2 g szl 1 1X 1X27 Jiwen He University of Houston Math 1432 F Section 26626 Lecture 20 March 27 2008 a mvmvev n xnk mm em mama Mam mm Mama mmquot a ammvmn Tax my name ammvmn Tax cm gamma

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