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# Calculus II MATH 1432

UH

GPA 3.69

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This 9 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1432 at University of Houston taught by Jiwen He in Fall. Since its upload, it has received 30 views. For similar materials see /class/208374/math-1432-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

LGCture 1 lsection 87 Numerical Integration J iwen He 1 Riemann Sums 11 Area Problem Area Problem 1 V I 9192 93 9n a b x a x0 x1 x2 x3 x b x Partition of a 9 Take a partition P 2 130561 xn of ab Then P splits up the interval a 9 into a nite number of subintervals 5130 131 58n17513n with a 80 lt 5131 lt lt Slim 2 b We have ab 51305131 U U cat451 U U CEn1CEn Remark This breaks up the region 9 into n subregions 21 Qn We can estimate the total area Off by estimating the area of each sabregion 21 and adding up the results 12 Lower and Upper Sums Lower and Upper Sums m I 1 I I I II l I l I I I l I l l I Let mfg 13141 mi 2 Qz luUQiUH UQn raftAsa 2 area of rz 3 area of 21 g div area of shaded region is a area of shaded region is an lower sum forf upper sum forf n b n i1 a i1 1 3 Riemann Sum Riemann Sum 12 y y v 1 X I 1 X2 X3 X4 X5 A X6 X7 8 l area of shaded region Is a 0 1 2 3 4 5 6 quot7 8 area of shaded region is an lower sum forf upper sum forf Riemann Sum mm fiA1 fv Av2 room where is any point picked in 1371i for z39 l n We have L141 Emmi g SP Z mam g ZMiAxi 2 Ufa i1 i1 i1 Limit of Riemann Sums b foe dx I 113180 room fv Av2 mama 2 Rectangle Approximations 2 1 Regular Partition Regular Partition A V Problem 2 d Find the approximate value of ln 2 f 1 It 6 7 8 9 1 th 1 f l t 1 2 uslngony evaueso fz ma 7 5 5 5 57 Numerical Integration Approximate f f d3 using only values of f at n 1 equally spaced points between a and 17 Regular Partition of a 17 Let 33 a l iALE 73 01 n Where A3 b Ta Then ab 130331 U U 3371 13 U 22 Approximations Rectangle Approximations X 1 W AL xil x X 26171 Xi X Xi1 Xi X f flta2gt da fltai1gtAa fezma f xquot 12xquotAa xi l Rectangle Approximations of fjil f d3 fezAme mam frrm leftendpoints rightendpoints midpoints Rectangle Approximations of f f d3 21 xi f d3 i 1 Efrem Zitagtva 2 Am i 1 1 7 21 left endpoints right endpoints midpoints Rectangle Rules b Rectangle Rule for Approximating fa d9 0 Left endpoint rule Ln b a f90 f91 fan1 b o nght endpomt rule Rn W1 fw2 mm o Midpoint rule Mn 7 b a 11 937171 937 n 2 quot39fTl y x Problem Find the approximate value of 1112 12 d 118ng Only the Values Of f l at l g g g 2 Approximating 1112 169314718 51s07456 RBf ltgg g gt 06456 M5 l g 02 gtm06919 23 Error Estimates Error Estimates 7 Nb 7777 gt Ti l I gt I I I W J I I I I I I I I Nb arm I I I i ii gt I I I I I I I Ha gt Pg u 7 r each width is Iquot here 2 6 b b a IImIdm Ln 3 n 1 fa Error Estimates b 1 b Left endpoint rule E5 f d3 Ln 6 fc 0Am a n b 1 b Right endpoint rule E5 f d3 Rn 6 fc 0Am a n M 7 b 7 i b 03 7 2 o Mldpomt rule En 7 d3 Mn 7 24 2 f c i 0Am a n Error Estimates Example Error Estimates b 1 b a2 o Left endpoint rule E7 d3 Ln 5 fc 0Aac a n b 1 b a2 Right endpoint rule E5 f d3 Rn 5 fc 0Am a n M 7 b 7 i b a 7 2 o Mldpomt rule En i d3 Mn 7 24 c 0Am a n A Approximating In 2 069314718 Note that 7 E127 fl 6 2 1 2 1E511E 1 lt 5 maXce12if Ci 1 10 2 1 3 S 52 maXCE12 LINCM W10 3 Trapeziodal and Parabolic Approximations 3 1 Approximations Trapeziodal and Parabolic Approximations V 7 f f f 9 I Xi1 quotIi 1 X x x1 x x xi l Xi x 2 f5quot 1 f d3 Trapeziodal Parabolic L Approximations of fig fda fi 1 fi Am fz 1 4f xi 12xi fag A51 trapezioa al Parabolic Approximations of f fa d3 2121 3 fz d3 2 1fltxi1gt 11 7 2121 Mail 4f 5 125 quot ma T rapezioa al Parabolic Trapeziodal and Simpson s Rules b Trapeziodal and Simpson s Rules for Approximating fa d9 0 Trapeziodal Rule Tn b2 na f930 2f931 2f93n71 b o Simpson7s Rule Parabolic Sn 6 na f9o 2 2fan1 wwwwwn y x Problem Find the approximate value of ln2 12 If using only the values of f i at 1 E I 2 2 7 57 57 57 57 39 Approximating ln2 T 069371816 10 REUF7 Jr Sse12lt2 10 10 10 10 10 MHETTH 32 Error Estimates Error Estimates 1717 fa P bia fb ifu biu each width is here 2 l1 f mw n Error Estimates b o Trapeziodal Rule E7 lt 2 1 b a n 6 elm Tn b o Simpsorfs Rule ES clan Sn Error Estimates Example Error Estimates b o Trapeziodal Rule E7 elm Tn b o Simpsorfs Rule ES clan Sn 12 1 b a5 2880 12 1 b a5 2880 gtU f 19 ommm 19 ommm f4c 0A4 f4c 0A4 x Approximating ln2 069314718 Note that faj m 123 f 9 7 f3L39 m 47 f493 271 IEEI s 1 1 52gt maXce12lfCl 271 5 IE S FEM 54 maXcem f40 Outline Contents 1 Riemann Sums 1 11 Area Problem 1 12 Lower and Upper Sums 1 13 Riemann Sum 2 2 Rectangle Approximations 2 21 Regular Partition 2 22 Approximations 3 23 Error Estimates 4 3 Trapeziodal and Parabolic Approximations 6 31 Approximations 6 32 Error Estimates 7

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