Simpsons 13 rd Rule of Integration Chemical Engineering

Simpson’s 1/3 rd Rule of Integration Chemical Engineering Majors Authors: Autar Kaw, Charlie Barker http: //numericalmethods. eng. usf. edu Transforming Numerical Methods Education for STEM Undergraduates 2/20/2021 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 In an attempt to understand the mechanism of the depolarization process in a fuel cell, an electro-kinetic model for mixed oxygen-methanol current on platinum was developed in the laboratory at FAMU. A very simplified model of the reaction developed suggests a functional relation in an integral form. To find the time required for 50% of the oxygen to be consumed, the time, T (s) is given by a) b) c) 11 Use Simpson’s 1/3 rd Rule to find the time required for 50% of the oxygen to be consumed. 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 is True Error 13 lmethods. eng. usf. edu http: //numerica

Solution (cont) c. ) 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 In an attempt to understand the mechanism of the depolarization process in a fuel cell, an electro-kinetic model for mixed oxygen-methanol current on platinum was developed in the laboratory at FAMU. A very simplified model of the reaction developed suggests a functional relation in an integral form. To find the time required for 50% of the oxygen to be consumed, the time, T (s) is given by a) b) c) 21 Use 4 -segment Simpson’s 1/3 rd Rule to find the time required for 50% of the oxygen to be consumed. 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. ) cont. 24 lmethods. eng. usf. edu http: //numerica

Solution (cont. ) b) c) 25 In this case, the true error is The absolute relative true error lmethods. eng. usf. edu http: //numerica

Solution (cont. ) Table. Values of Simpson’s 1/3 rd Rule for Example 2 with multiple segments Approximate Value 2 4 6 8 10 26 190160 190140 − 24. 100 − 1. 9838 0. 42010 0. 13655 0. 056663 0. 012675% 0. 0010434% 2. 2094× 10− 4% 7. 1815× 10− 5% 2. 9802× 10− 5% 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 27 lmethods. eng. usf. edu http: //numerica

Error in the Multiple Segment Simpson’s 1/3 rd Rule . . . 28 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 29 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 30 lmethods. eng. usf. edu http: //numerica

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