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# Class Note for MATH 1432 at UH 2

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This 10 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 19 views.

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Date Created: 02/06/15

Lecture 20ection 107 Improper Integrals Jiwen He 1 Improper Integrals What are Improper Integrals 1 1 1 jazz 741 1 I 0 I b b b Knownfzdz i2dzil 17 0ltaltb7 a a z a b I a o the interval of integration db7 0 lt a lt b7 is bounded7 1 o the function being integrated 072 is bounded over db By a limit process7 we can extend the integration process to 00 b o unbounded intervals elgi7 loo i2 dz lim i dz lim 1 z bgtoo 1 12 bgtoo l l 1 o unbounded functions elgi7 as I A 07 12 A 00 ide 0 z 1 lim ide lim 00 z agt0 a agt0 a l 11 Integrals Over Unbounded Intervals Integrals Over Unbounded Intervals J A a b X 00 I Let f be continuous on 1 We de ne05ex dx blim dx The improper integral converges if the limit exists The improper integral di verges if the limit doesn t exist If blim y 0 then dx diverges b b If f is continuous on 700 dx lim dx If f is cont aaioo a on 700 00 dx xd dx Fundamental Theorem of Integral Calculus Let Fx for V36 2 1 Then 00 b a mom b11330 mom gym b13130 FltbgteFltagt FM xlgngoFWFFW Let F for Vac g b Then I b fxdxFx Fbi lim 00 maioo Let F for Vac Then 2xdx0 xdxfxdxFxo lim Fxi lim Fee 12an maioo no 7 lim Fm hm Fm F0gt maioo zaoo Examples m 2 2 1Holtgtlt2gt2 ldz lnzlo lim lnz7lnl 00 1 zgtoo oo 721 C 72z 721 776 7 ie 7 fl fl A6 2021324 2 2 00 DO 2 C 2 dz tan 1 imcz C 700 What is Wrong with This What is wrong with the following argument 1 2 00 b 7 2z 7 z 7 2 b 2z 7 007 007112d17bl1i1010 7b7112d17bligt11010ltl1m11 Nib a blggoibiljr dz70 21 dglim lnlb2 ilnl 7122 lirn00lnlz2 mlimlnlz2 ilnl b b 0 0 H00 HOG zgtoo lz2 00 b 0 Note that we did not de ne dz as blim dzl oo 700 9quot 7b dz L blim dz L 700 400 7b b 00 blim L b L 00 7b 700 b o For every odd function f7 we have blim dz 0 but this is ADC 75 certainly not the case for dzl flooxiadz7agt0 xquot 1 001 7 dx a1 1 350 007 1fozgl Ifa7 17 then dx xlio hm xlia 1 350 l a 1 l azaoo l a 007 ifa lt1 If a 17 then Amidx lnx o 111x h11 oo Quiz Quiz 2 1 Find hm 17gtoo em a 07 b 17 c 00 2 Find lim 1 zgtoo 111 t a 07 b 17 c 00 12 Integrals of Unbounded Functions Integra 39 YA I R Let f be continuous on 1 Q but gt ioocas a gt b We de ne f 9 d f93 d9 lim 9 The improper integral cJ eryes if therhrmtems s The improper integral di verges if the limit doesn7t exist If f is continuous on 1 bbbut gt too as a gt 1 f9d9 lim d If f is cont on 1 b e eept at somecjw f Ce in a gt e 07 a gt 0 fab wm wdmf xmw Fundamental Theorem of Integral Calculus Let Faj for V9 6 1 b and gt too as a gt 13 Then b Ceb ceb b c d9 Clinr d9 lim lim Fe F1 1 b If lim d200 then f d9 diverges zeb Let Faj for V9 6 1 b and gt too as a gt 1 Then I fada Fa Fb lim Fm mHaJr a 11 Where gt too as Examples 1 EHO t 11 32 3 3 3 7 13 23 Azude 2x 2 hm 1 1 2 0 1 1 1 7d 1 1171 1 0 I z nz10 n Egg nz 00 7r tanzdz 1nsecz107 11m 1nsecz71n1 00 0 EH Examples 00 671 0 idu 1 1 7d 7 id A 1176 2 I 1 xliuQ A xliuQ u 1 1 7r s1n 1110 5 Set u 6 then du 76 dz and u2 6 2 3 sinz dziOi duil 1 0 1200s171 7 1 72 0 1 1 1W11 du Set u Zoosz 7 17 then du 72sinzdz1 What is Wrong with This What is wrong With the fo110Wing argument 4 4 1 1 3 7d 7 7 0 11722 I 1721 2 2 1 1 2 1 id 77 139 7 1 11722 I 1721 xLIE lt 172 00 y 1 3 4 x 4 1 Note that W gt 00 as 2 gt 2 or 2 gt 2 To evaluate 2 dz we 1 a 2 2 1 4 1 need to calculate the impTopeT integrals 1 a 7 22 dzz and 2 a 7 22 dx 01 1 dx 04 gt O If a 17 then If a 17 then Quiz Quiz 1 11 7 ifozlt17 dx 1 6 0 95a 007 1f0421 1 1 1 1 1 1 dx xlio limx 350 1 a 0 l a 1 Clzgt0 i 7 ifozlt17 i 007 ifagt1 11 015 dx lnx lnl 11m 111 00 gtO 1 3 Find 11m xsin zgtoo x a 07 b 17 c 00 4 Find lim 1 17gt0 em a 07 b 17 c 00 xquot 2 Comparison Test for Convergence 21 Comparison Test for Convergence y I gt x Let be improper integration over an interval I7 and suppose that f and g are I continuous on I such that 0 me gym w e I o Ifgz dz czmvezges7 then so does dzi I I o Iffz dz diverges7 then so does dzi I I oo 1 The im r0 er inte ral 7 dz conver es since quot p g 1 W 9 0lt 1 lt 1 V gt1 d m 1 d 7 7 7 z 7 an 7 z converges 1z3 xz3 1 z3 1 The im r0 er inte ral 7 dz diver es since quot p g 1 W 9 0 lt 1 1 V gt1 d m 1 d d39 71IltW zi an 1 11 z werges Out line Contents 1 Improper Integrals 1 1 Unbounded Intervals t t t t t t t t t t t t t t t t t t t t t t t t 1 2 Unbounded Functions 2 Comparison Test 21 Comparison Test t t t t t t t t t t t t t t t t t t t t t t t t t t t 10

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