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TAMU PETE 301 - Numerical Methods for Engineers Ch. 21 Solutions

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1 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. CHAPTER 21 21.1 (a) Analytical solution:  /2 /2 0 0 (6 3cos ) 6 3sin 6( /2) 3sin( /2) 0 12.42478xdx x x (b) Trapezoidal rule (n = 1): 96(1.570796 0) 11.780972I 12.42478 11.78097100% 5.182%12.42478t (c) Trapezoidal rule (n = 2): 9 2(8.12132) 6(1.570796 0) 12.26896 1.254%4tI   Trapezoidal rule (n = 4): 9 2(8.771639 8.12132 7.14805) 6(1.570796 0) 12.38613 0.311%8tI   (d) Simpson’s 1/3 rule: 9 4(8.12132) 6(1.570796 0) 12.43162 0.055%6tI   (e) Simpson’s rule (n = 4): 9 4(8.771639 7.14805) 2(8.12132) 6(1.570796 0) 12.42518 0.0032%12tI   (f) Simpson’s 3/8 rule: 9 3(8.598076 7.5) 6(1.570796 0) 12.42779 0.024%8tI   (g) Simpson’s rules (n = 5): 9 4(8.85317) 8.427051(0.628319 0)68.427051 3(7.763356 6.927051) 6 (1.570796 0.628319)8 5.533364 6.891665 12.42503 I 0.002%t 21.2 (a) Analytical solution:  3 32 2 2(3) 2(0) 0 0 1 0.5 3 0.5 0 0.5 2.501239xxedxx e e e  (b) Trapezoidal rule (n = 1): 0 0.997521(3 0) 1.4962822I  2.501239 1.496282100% 40.18%2.501239t (c) Trapezoidal rule (n = 2): 0 2(0.950213) 0.997521(3 0) 2.17346 13.10%4tI   Trapezoidal rule (n = 4):2 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. 0 2(0.77687 0.950213 0.988891) 0.997521(3 0) 2.411051 3.61%8tI   (d) Simpson’s 1/3 rule: 0 4(0.950213) 0.997521(3 0) 2.399186 4.08%6tI   (e) Simpson’s rule (n = 4): 0 4(0.77687 0.988891) 2(0.950213) 0.997521(3 0) 2.490248 0.44%12tI    (f) Simpson’s 3/8 rule: 0 3(0.864665 0.981684) 0.997521(3 0) 2.451213 2.00%8tI   (g) Simpson’s rules (n = 5): 0 4(0.698806) 0.909282(1.2 0)60.909282 3(0.972676 0.99177) 0.997521 (3 1.2)8 0.740901 1.755032 2.495933 0.21%tI  21.3 (a) Analytical solution: 426 435 4 2 2 (1 4 2 ) 110423xxxx xdxx x       (b) Trapezoidal rule (n = 1): 29 1789(4 ( 2)) 52802I  1104 5280100% 378.26%1104t (c) Trapezoidal rule (n = 2): 29 2( 2) 1789(4 ( 2)) 2634 138.59%4tI   Trapezoidal rule (n = 4): 29 2(1.9375 2 131.3125) 1789(4 ( 2)) 1516.875 37.398%8tI     (d) Simpson’s 1/3 rule (n = 2): 29 4( 2) 1789(4 ( 2)) 1752 58.7%6tI   (e) Simpson’s 3/8 rule: 29 3(1 31) 1789(4 ( 2)) 1392 26.087%8tI     (f) Boole’s rule (n = 5):3 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. 7( 29) 32(1.9375) 12( 2) 32(131.3125) 7(1789)(4 ( 2)) 1104 0%90tI     21.4 Analytical solution: 23 22 1 14 ( 2 ) 4 8.333333xx/xdx xx Trapezoidal rule (n = 1): 99(2 1) 92 The results are summarized below: n Integral t 1 9 8.00% 2 8.513889 2.17% 3 8.415185 0.98% 4 8.379725 0.56% 21.5 Analytical solution: 5 534 3 31 (4 3) (4 3) 205616xdx x   Simpson’s rule (n = 4): 3375 4( 343 729) 2(1) 4913(5 ( 3)) 2056 0%12tI   Simpson’s rules (n = 5): 3375 4( 636.056) 10.648(0.2 ( 3))610.648 3(74.088 1191.016) 4913 (5 0.2) 3162.6 5218.598 2056 0%8tI        Because Simpson’s rules are perfect for cubics, both versions yield the exact result for this cubic polynomial. 21.6 Analytical solution: 3 322 0 0 ( 2 2) 98.42768xxxe dx x x e  Trapezoidal rule (n = 4): 0 2(1.190813 10.0838 48.03166) 180.7698(3 0) 112.2684 14.062%8tI   Simpson’s rule (n = 4): 0 4(1.190813 48.03166) 2(10.0838) 180.7698(3 0) 99.45683 1.046%12tI   21.7 Analytical solution: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. 1 122 0 01 14 14 36.945012ln14xxdx (a) Trapezoidal rule (n = 1): 1196(1 0) 98.52I  36.94501 98.5100% 166.612%36.94501t (b) Simpson’s 1/3 rule (n = 2): 1 4(14) 196(1 0) 42.16667 14.134%6tI   (c) Simpson’s 3/8 rule: 1 3(5.808786 33.74199) 196(1 0) 39.45654 6.798%8tI   (d) Boole’s rule: 7(1) 32(3.741657) 12(14) 32(52.3832) 7(196)(1 0) 37.14439 0.5397%90tI   (e) Midpoint method: (1 0)14 14 62.106%tI   (f) 3-segment-2-point open integration


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TAMU PETE 301 - Numerical Methods for Engineers Ch. 21 Solutions

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