1 CHAPTER 21 21 1 a Analytical solution 2 0 2 6 3cos x dx 6 x 3sin x 0 6 2 3sin 2 0 12 42478 b Trapezoidal rule n 1 9 6 12 42478 11 78097 I 1 570796 0 11 78097 t 100 5 182 2 12 42478 c Trapezoidal rule n 2 9 2 8 12132 6 12 26896 t 1 254 I 1 570796 0 4 Trapezoidal rule n 4 9 2 8 771639 8 12132 7 14805 6 I 1 570796 0 12 38613 t 0 311 8 d Simpson s 1 3 rule 9 4 8 12132 6 12 43162 t 0 055 I 1 570796 0 6 e Simpson s rule n 4 9 4 8 771639 7 14805 2 8 12132 6 12 42518 t 0 0032 I 1 570796 0 12 f Simpson s 3 8 rule 9 3 8 598076 7 5 6 12 42779 t 0 024 I 1 570796 0 8 g Simpson s rules n 5 9 4 8 85317 8 427051 I 0 628319 0 6 8 427051 3 7 763356 6 927051 6 1 570796 0 628319 8 5 533364 6 891665 12 42503 t 0 002 21 2 a Analytical solution 1 e dx x 0 5e 3 2 x 0 2 x 3 0 3 0 5e 2 3 0 0 5e 2 0 2 501239 b Trapezoidal rule n 1 0 0 997521 I 3 0 1 496282 2 2 501239 1 496282 t 100 40 18 2 501239 c Trapezoidal rule n 2 0 2 0 950213 0 997521 2 17346 t 13 10 I 3 0 4 Trapezoidal rule n 4 PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 2 I 3 0 0 2 0 77687 0 950213 0 988891 0 997521 2 411051 8 t 3 61 d Simpson s 1 3 rule 0 4 0 950213 0 997521 2 399186 t 4 08 I 3 0 6 e Simpson s rule n 4 0 4 0 77687 0 988891 2 0 950213 0 997521 2 490248 I 3 0 12 t 0 44 f Simpson s 3 8 rule 0 3 0 864665 0 981684 0 997521 2 451213 t 2 00 I 3 0 8 g Simpson s rules n 5 0 4 0 698806 0 909282 I 1 2 0 6 0 909282 3 0 972676 0 99177 0 997521 3 1 2 8 0 740901 1 755032 2 495933 t 0 21 21 3 a Analytical solution 4 2 x2 x6 1 x 4 x 2x dx x x4 2 3 3 4 1104 5 b Trapezoidal rule n 1 29 1789 I 4 2 5280 2 c Trapezoidal rule n 2 29 2 2 1789 I 4 2 2634 4 2 t 1104 5280 100 378 26 1104 t 138 59 Trapezoidal rule n 4 29 2 1 9375 2 131 3125 1789 I 4 2 1516 875 8 d Simpson s 1 3 rule n 2 29 4 2 1789 I 4 2 1752 6 e Simpson s 3 8 rule 29 3 1 31 1789 I 4 2 1392 8 t 37 398 t 58 7 t 26 087 f Boole s rule n 5 PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 3 I 4 2 7 29 32 1 9375 12 2 32 131 3125 7 1789 1104 90 t 0 21 4 Analytical solution 2 1 x3 4 x 2 x dx 4 x x 3 2 8 33333 2 1 Trapezoidal rule n 1 9 9 2 1 9 2 The results are summarized below n 1 2 3 4 Integral 9 8 513889 8 415185 8 379725 t 8 00 2 17 0 98 0 56 21 5 Analytical solution 5 3 1 4 x 3 dx 4 x 3 4 16 5 2056 3 3 Simpson s rule n 4 3375 4 343 729 2 1 4913 I 5 3 2056 12 t 0 Simpson s rules n 5 3375 4 636 056 10 648 I 0 2 3 6 10 648 3 74 088 1191 016 4913 t 0 5 0 2 3162 6 5218 598 2056 8 Because Simpson s rules are perfect for cubics both versions yield the exact result for this cubic polynomial 21 6 Analytical solution 3 0 x 2 e x dx x 2 2 x 2 e x 3 0 98 42768 Trapezoidal rule n 4 0 2 1 190813 10 0838 48 03166 180 7698 I 3 0 112 2684 8 Simpson s rule n 4 0 4 1 190813 48 03166 2 10 0838 180 7698 I 3 0 99 45683 12 t 14 062 t 1 046 21 7 Analytical solution PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 4 1 0 1 1 142 x dx 142 x 36 94501 2 ln14 0 a Trapezoidal rule n 1 1 196 I 1 0 98 5 2 t b Simpson s 1 3 rule n 2 1 4 14 196 I 1 0 42 16667 6 36 94501 98 5 100 166 612 36 94501 t 14 134 c Simpson s 3 8 rule 1 3 5 808786 33 74199 196 I 1 0 39 45654 t 6 798 8 d Boole s rule 7 1 32 3 741657 12 14 32 52 3832 7 196 I 1 0 37 14439 t 0 5397 90 e Midpoint method I 1 0 14 14 t 62 106 f 3 segment 2 point open integration formula 5 808786 33 74199 I 1 0 t 46 473 19 77539 2 g 4 segment 3 point open integration formula 2 3 741657 14 2 52 3832 I 1 0 32 74991 3 t 11 355 21 8 Analytical solution 3 0 5 3cos x dx 5 x 3sin x 0 15 42336 3 a Trapezoidal rule n 1 8 2 030023 I 3 0 15 04503 2 b Simpson s 1 3 rule n 2 8 4 5 212212 2 030023 I 3 0 15 …
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