Simpsons 13 rd Rule of Integration Civil Engineering
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Simpson’s 1/3 rd Rule of Integration Civil Engineering Majors Authors: Autar Kaw, Charlie Barker http: //numericalmethods. eng. usf. edu Transforming Numerical Methods Education for STEM Undergraduates 9/17/2021 http: //numericalmethods. eng. usf. edu 1
rd 1/3 Simpson’s Rule of Integration http: //numericalmethods. eng. usf. edu
What is Integration? Integration The process of measuring the area under a curve. f(x) y Where: f(x) is the integrand a= lower limit of integration b= upper limit of integration 3 a b lmethods. eng. usf. edu x http: //numerica
Simpson’s 1/3 rd Rule 4 lmethods. eng. usf. edu http: //numerica
Basis of Simpson’s 1/3 rd Rule Trapezoidal rule was based on approximating the integrand by a first order polynomial, and then integrating the polynomial in the interval of integration. Simpson’s 1/3 rd rule is an extension of Trapezoidal rule where the integrand is approximated by a second order polynomial. Hence Where 5 is a second order polynomial. lmethods. eng. usf. edu http: //numerica
Basis of Simpson’s 1/3 rd Rule Choose and as the three points of the function to evaluate a 0, a 1 and a 2. 6 lmethods. eng. usf. edu http: //numerica
Basis of Simpson’s 1/3 rd Rule Solving the previous equations for a 0, a 1 and a 2 give 7 lmethods. eng. usf. edu http: //numerica
Basis of Simpson’s 1/3 rd Rule Then 8 lmethods. eng. usf. edu http: //numerica
Basis of Simpson’s 1/3 rd Rule Substituting values of a 0, a 1, a 2 give Since for Simpson’s 1/3 rd Rule, the interval [a, b] is broken into 2 segments, the segment width 9 lmethods. eng. usf. edu http: //numerica
Basis of Simpson’s 1/3 rd Rule Hence Because the above form has 1/3 in its formula, it is called Simpson’s 1/3 rd Rule. 10 lmethods. eng. usf. edu http: //numerica
Example 1 The concentration of benzene at a critical location is given by where So in the above formula Since a) b) c) 11 decays rapidly as , we will approximate Use Simpson 1/3 rd rule to find the approximate value of erfc(0. 6560). Find the true error, for part (a). Find the absolute relative true error, for part (a). lmethods. eng. usf. edu http: //numerica
Solution a) 12 lmethods. eng. usf. edu http: //numerica
Solution (cont) b) The exact value of the above integral cannot be found. We assume the value obtained by adaptive numerical integration using Maple as the exact value for calculating the true error and relative true error. True Error 13 lmethods. eng. usf. edu http: //numerica
Solution (cont) c) The absolute relative true error, 14 lmethods. eng. usf. edu http: //numerica
Multiple Segment Simpson’s 1/3 rd Rule 15 lmethods. eng. usf. edu http: //numerica
Multiple Segment Simpson’s 1/3 rd Rule Just like in multiple segment Trapezoidal Rule, one can subdivide the interval [a, b] into n segments and apply Simpson’s 1/3 rd Rule repeatedly over every two segments. Note that n needs to be even. Divide interval [a, b] into equal segments, hence the segment width where 16 lmethods. eng. usf. edu http: //numerica
Multiple Segment Simpson’s 1/3 rd Rule f(x) . . . x x 0 x 2 xn-2 xn Apply Simpson’s 1/3 rd Rule over each interval, 17 lmethods. eng. usf. edu http: //numerica
Multiple Segment Simpson’s 1/3 rd Rule Since 18 lmethods. eng. usf. edu http: //numerica
Multiple Segment Simpson’s 1/3 rd Rule Then 19 lmethods. eng. usf. edu http: //numerica
Multiple Segment Simpson’s 1/3 rd Rule 20 lmethods. eng. usf. edu http: //numerica
Example 2 The concentration of benzene at a critical location is given by where So in the above formula Since a) b) c) 21 decays rapidly as , we will approximate Use four segment Simpson’s 1/3 rd Rule to find the approximate value of erfc(0. 6560). Find the true error, for part (a). Find the absolute relative true error, for part (a). lmethods. eng. usf. edu http: //numerica
Solution a) Using n segment Simpson’s 1/3 rd Rule, So 22 lmethods. eng. usf. edu http: //numerica
Solution (cont. ) 23 lmethods. eng. usf. edu http: //numerica
Solution (cont. ) b) c) 24 In this case, the true error is The absolute relative true error lmethods. eng. usf. edu http: //numerica
Solution (cont. ) Table 1: Values of Simpson’s 1/3 rd Rule for Example 2 with multiple segments Approximate Value 2 4 6 8 10 25 − 0. 47178 − 0. 30529 − 0. 30678 − 0. 31110 − 0. 31248 0. 15846 − 0. 0080347 − 0. 0065444 − 0. 0022249 − 0. 00084868 50. 573% 2. 5643% 2. 0887% 0. 71009% 0. 27086% lmethods. eng. usf. edu http: //numerica
Error in the Multiple Segment Simpson’s 1/3 rd Rule The true error in a single application of Simpson’s 1/3 rd Rule is given as In Multiple Segment Simpson’s 1/3 rd Rule, the error is the sum of the errors in each application of Simpson’s 1/3 rd Rule. The error in n segment Simpson’s 1/3 rd Rule is given by 26 lmethods. eng. usf. edu http: //numerica
Error in the Multiple Segment Simpson’s 1/3 rd Rule . . . 27 lmethods. eng. usf. edu http: //numerica
Error in the Multiple Segment Simpson’s 1/3 rd Rule Hence, the total error in Multiple Segment Simpson’s 1/3 rd Rule is 28 lmethods. eng. usf. edu http: //numerica
Error in the Multiple Segment Simpson’s 1/3 rd Rule The term is an approximate average value of Hence where 29 lmethods. eng. usf. edu http: //numerica
Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, Math. Cad and MAPLE, blogs, related physical problems, please visit http: //numericalmethods. eng. usf. edu/topics/simpsons_ 13 rd_rule. html
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