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# Note 11 for MATH 1432 with Professor He at UH

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This 9 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 21 views.

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Date Created: 02/06/15
LGCture 1 lsection 87 Numerical Integration J iwen He 1 Riemann Sums 11 Area Problem Area Problem 91522 Q3 914 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 Q 21 U U 21 U U 9n 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 I l 1 er I l I I 1 l J l l l l I I X a x 1 b a x 1 x b a n gt1 v Let A5132 CCi CCZ391 mi 2 min fI Mi 2 max ft raftAsa 2 area of rz 3 area of 21 S QCElwi lafI z l welwi lawi 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 v y 39 y V I X2 X3 X4 X6 39739 8 X area of shaded region Is a 0 1 x2 3 quot4 5 6 quot 7 quot398 area of shaded region is an lower sum forf upper sum forf 1 39 gte Riemann Sum me fv1 Aa1 room 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 Problem Find the approximate value of 1 using only the values of f at Numerical Integration M 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 32 a l z Aaj 73 O1 n Where A3 b Ta Then 22 Approximations Rectangle Approximations f k xi l Ni x xiil xi X Xi l Xi X fail 33 d3 f 13i 1A 13 f1ZAz Rectangle Approximations of fjil f d3 lm HAZE mam f W Ax leftendpoints rightendpoints midpoints Rectangle Approximations of f f d3 21 3 f d3 fiilAfL397 ffL39iAL397 Ag left endpoints right endpoints midpoints Rectangle Rules b Rectangle Rule for Approximating fa d9 b a o Left endpomt rule Ln o Right endpoint rule Rn b fw1 fw2 f93nl o Midpointrule Mn ba f930931 fn71n n 2 2 y Problem dm Find the approximate value of 1112 If at 1 6 7 8 2 using only the values of f i 7373va 572 Approximating 1112 169314718 L51 07456 R5 06456 Mslti21010 23 Error Estimates Error Estimates f v 77 77 fb u Ha biz II each width is here 2 6 imam Ln b fb fa Error Estimates b 2 Left endpoint rule E5 f d3 Ln ab 6 fc 0Am a TL b 1 b a2 Right endpoint rule E5 f d3 Rn 5 fc 0Am a TL M 7 b 7 i b a 7 2 o Mldpomt rule En i d3 Mn 7 24 2 f c 0Am TL Error Estimates Example Error Estimates b 2 Left endpoint rule E5 f d3 Ln ab 6 fc 0Am a TL b 1 b a2 Right endpoint rule E5 f d3 Rn 5 fc 0Am a TL M 7 b 7 i b 03 7 2 o Mldpomt rule En i d3 Mn 7 24 2 f c 0Am TL 11 mm U1I U39IIOC who J Approximating In 2 069314718 Note that 7 E127 fl 6 2 1 2 1E 11E 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 1141 X x x 1 x x xi l xi x 2 f 1 1 f d3 Trapeziodal Parabolic z Approximations of fig fda 1fltmgt fltscigt1 Ax f337 1 4f ag A51 trapezioa al Parabolic Approximations of f fa d3 2121 fail fz d3 2121 iflt33i 1 a 2121 f33i 1 4f 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 o Simpson7s Rule Parabolic Sn a form fwn 2fltw1gt 210mm wwwwwn Problem Find the approximate value of ln2 If If using only the values of f i 6 7 8 9 at173737g7372 Approximating ln2 T 069371816 10 REUF7 32 Error Estimates Error Estimates f I f a fb u biz II each width is here 2 6 TL fabmdm Tn C 1 fa Error Estimates b 3 1 b Trapeziodal Rule E5 108 cm Tn E 0 f c i OA2 a TL 5 5 Simpsorfs Rule ES 108 cm Sn imflt4gtc OA4 a 2880 n4 Error Estimates Example Error Estimates T 7 b i i b 603 2 o Trapemodal Rule En 7 d3 Tn i 12 2 f c 0Am a TL b 5 Simpsorfs Rule ES 108 cm Sn i b 6 flt4gtc OA4 a 2880 n4 fx Approximating ln2 069314718 Note that faj m 12 quot93 7 f3L39 m 47 f493 271 3 i S 52gt maXcE1721fCl F10 7 5 lE ll S Tlsoas j maxce12f4cl 39 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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