Using composite Trapezoidal and Simpson’s 1/3 rule, approximate Z p 0 xSin(x)dx. Obtain your results for n = 2, 4, 6, ...., 40 panels. For each case, calculate the error in the approximation using the...

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Using composite Trapezoidal and Simpson’s 1/3 rule, approximate Z p 0 xSin(x)dx. Obtain your results for n = 2, 4, 6, ...., 40 panels. For each case, calculate the error in the approximation using the exact value of the integral. For each n (number of panels), tabulate h (mesh size), result obtained with the trapezoidal method, error of the trapezoidal method, result obtained with Simpson’s method, and the error of the Simpson’s method. On the same log-log plot, show the change of the error with the mesh size for each method. Comment on your results. 2. Use 3 and 4 point Gaussian Quadrature to approximate the integral given in question 1. 3. To numerically evaluate Z 0.5 0 Z 0.5 0 xye(y-x) dydx (a) Use Simpson’s 1/3 method with two panels in each direction (b) Use 3 point Gaussian quadrature for each direction Show your calculations clearly and evaluate the error for each method using the exact value of the integral. 4. Following integral arises in the study of dihedral angle for assuring the lateral stability of an aircraft: Z 1 0 p (1 - ? 2 ) ? 2 d? (a) Numerically integrate the above integral using: (i) Simpson’s method with two panels, (ii) three-point Gaussian Quadrature. Compare your results with the exact value of the integral.


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AE/ME 330 Spring 2013, Homework VII, Due Monday April 29 1. Using composite Trapezoidal and Simpson's 1/3 rule, approximate Z  xSin(x)dx: 0 Obtain your results for n = 2; 4; 6;::::; 40 panels. For each case, calcu- late the error in the approximation using the exact value of the integral. For each n (number of panels), tabulate h (mesh size), result obtained with the trapezoidal method, error of the trapezoidal method, result ob- tained with Simpson's method, and the error of the Simpson's method. On the same log-log plot, show the change of the error with the mesh size for each method. Comment on your results. 2. Use 3 and 4 point Gaussian Quadrature to approximate the integral given in question 1. 3. To numerically evaluate Z Z 0:5 0:5 (yx) xye dydx 0 0 (a) Use Simpson's 1/3 method with two panels in each direction (b) Use 3 point Gaussian quadrature for each direction Show your calculations clearly and evaluate the error for each method using the exact value of the integral. 4. Following integral arises in the study of dihedral angle for assuring the lateral stability of an aircraft: Z 1 p 2 2 (1 )  d 0 (a) Numerically integrate the above integral using: (i) Simpson's method with two panels, (ii) three-point Gaussian Quadrature. Compare your results with the exact value of the integral. 1(b) Another three-point quadrature rule for this type of integrals can be written as Z 1 p 2 (1 ) f()d! f(0) +! f(0:5) +! f(1) 0 1 2 0 which is exact for quadratic polynomials (i.e., above formula will 2 be exact for f() = 1, f() =, and f() = ). Using this fact, determine the weight coecients ! , ! , and ! . 0 1 2 2



Answered Same DayDec 22, 2021

Answer To: Using composite Trapezoidal and Simpson’s 1/3 rule, approximate Z p 0 xSin(x)dx. Obtain your results...

Robert answered on Dec 22 2021
127 Votes
1) ∫



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=3.14
Trapezoidal rule for n =2



(




)
For n =3



(








)
For n =4

For n =5

For n =6

For n =7

For n =8

n Approximate value Error
9 3.11 0.03
10 3.116 0.024
11 3.12 0.02
12 3.124 0.016
13 3.126 0.014
14 3.128 0.012
15 3.13 0.1
16 3.131 0.009
17 3.133 0.007
18 3.134 0.006
19 3.134 0.006
20 3.135 0.005
21 3.136 0.004
22 3.136 0.004
23 3.137 0.003
24 3.137 0.003
25 3.137 0.003
26 3.138 0.002
27 3.138 0.002
28 3.138 0.002
29 3.13852 0.00148
30 3.13872 0.00128
31 3.1389 0.0011
32 3.13907 0.00093
33 3.13922 0.00078
34 3.13936 0.00064
35 3.13948 0.00052
36 3.13960 0.00040
37 3.13971 0.00027
38 3.13980 0.00020
39 3.13989 0.00011
40 3.13998 0.00002
Simpson 1/3 rule
n=2



(




)

n Approximate Value Error
2 3.289868134 -0.149868134
3 2.53254167 0.60745833
4 3.1487551 -0.0087551
5 2.930657603 0.209342397
6 3.142948549 -0.002948549
7 3.03504795 0.10495205
8 3.142015465 -0.002015465
9 3.077400861 0.062599139
10 3.141764683 -0.001764683
11 3.098708854 0.041291146
12 3.141675316 -0.001675316
13 3.110924725 0.029075275
14 3.141637176 -0.001637176
15 3.118574474 0.021425526
16 3.141618715 -0.001618715
17 3.12368067 0.01631933
18 3.141608908 -0.001608908
19 3.127258045 0.012741955
20 3.141603311 -0.001603311
21 3.129861346 0.010138654
22 3.141599929 -0.001599929
23 3.131814693 0.008185307
24 3.141597788 -0.001597788
25 3.133317776 0.006682224
26 3.14159638 -0.00159638
27 3.13449908...
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