ENGINEERING MATHEMATICS 4 (BDA34003)

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1 ENGINEERING MATHEMATICS 4 (BDA34003) Lecture Module 6: Numerical Integration Waluyo Adi Siswanto Universiti Tun Hussein Onn Malaysia This work is licensed under a Creative Commons Attribution 3.0 License.

2 Topics Rectangular rule Trapezoidal rule Simpson's rule Simpson 1/3 Simpson 3/8 Gauss Quadrature 2-point 3-point Lecture Module 6 BDA

3 Learning Outcomes Students have the knowledge of effective numerical integration in engineering Students will be able to use various methods of numerical integrations to calculate complex problems with high accuracy without involving mathematical exact solution Students will be able to decide the appropriate numerical method to solve particular engineering problem Lecture Module 6 BDA

4 y y(x) You want to calculate the yellow area, under the curve y(x) Then you can calculate the area x n Area= x 1 y(x)dx x 1 x n x Lecture Module 6 BDA

5 y y 2 y n 1 y(x) If the function is difficult You will find it difficult to integrate. Then you can predict by approximation Area=Area 1+ Area 2+ Area 3 Area=h y 1 + h y 2 + h y n 1 y 1 Or you can write Area=h( y 1 + y 2 + y n 1 ) x 1 x n Of course not accurate. If you use smaller division? x Lecture Module 6 BDA

6 Rectangular rule y n 1 y 2 y y(x) If you want to divide by N division h= x n x 1 N Number of data n = N + 1 y 1 x n x 1 y(x)dx = h( y 1 + y y n 1) h h h x x 1 x n Lecture Module 6 BDA

7 Trapezoidal rule y n y n 1 y 3 y 2 y y(x) If you want to divide by N division h= x n x 1 N Now the area is not rectangular but trapezoidal y 1 x n y(x)dx = h x 1 2 ( y + y )+ h ( y + y )+ + h ( y + y ) n 1 n h h h x n y(x)dx = h x 1 2 ( y + 2 y y + y ) 1 2 n 1 n x x 1 x n Lecture Module 6 BDA

8 Trapezoidal m function (trapezoidal.m): function I = trapezoidal(f_str, a, b, n) %TRAPEZOIDAL Trapezoidal Rule integration. % I = TRAPEZOIDAL(F_STR, A, B, N) returns the Trapezoidal Rule approximation % for the integral of f(x) from x=a to x=b, using N divisions (subintervals), where % F_STR is the string representation of f. I=0; g = inline(f_str); h = (b-a)/n; I = I + g(a); for ii = (a+h):h:(b-h) I = I + 2*g(ii); end I = I + g(b); I = I*h/2; Lecture Module 6 BDA

9 Simpson's rule Simpson 1/3 x n y( x)dx = h x 1 3 ( y 1+ 4 y y y y y n 1+ y n ) The number of division must be multiplication of 2 Simpson 3/8 x n x 1 y( x)dx = 3 h 8 ( y 1+ 3( y 2 + y 3 + y 5 + y 6 + )+ 2( y 4 + y 7 + )+ y n ) The number of division must be multiplication of 3 Lecture Module 6 BDA

10 Simpson 1/3 m function (simpson13.m): function I = simpson13(f_str, a, b, n) %SIMPRULE Simpson's rule integration. % I = SIMPRULE(F_STR, A, B, N) returns the Simpson's rule approximation % for the integral of f(x) from x=a to x=b, using N subintervals, where % F_STR is the string representation of f. % An error is generated if N is not a positive, even integer. I=0; g = inline(f_str); h = (b-a)/n; if((n > 0) && (rem(n,2) == 0)) I = I + g(a); for ii = (a+h):2*h:(b-h) I = I + 4*g(ii); end for kk = (a+2*h):2*h:(b-2*h) I = I + 2*g(kk); end I = I + g(b); I = I*h/3; else disp('incorrect Value for N') end Lecture Module 6 BDA

11 Simpson 3/8 m function (simpson38.m): function int=simpson38(f_str,x1,x2,n) h=(x2-x1)/n; x(1)=x1; f=inline(f_str); if((n > 0) && (rem(n,3) == 0)) sum=f(x1); for i=2:n x(i)=x(i-1)+h; end for j=2:3:n sum=sum+3*f(x(j)); end for k=3:3:n sum=sum+3*f(x(k)); end for l=4:3:n sum=sum+2*f(x(l)); end sum=sum+f(x2); int=sum*3*h/8; else disp('incorrect Value for N') end Lecture Module 6 BDA

12 Example 6-1 Find the approximate value of 4 1 x x+ 1 dx By using : a) Exact integration, use SMath b) Rectangular rule, 12 division c) Trapezoidal rule, 12 division d) Simpson 1/3, 12 division e) Simpson 3/8, 12 division Lecture Module 6 BDA

13 Exact integration in SMath Rectangular rule N = 12 h = 0.25 No data x x/sqrt(x+1) Result=0.25( )= Lecture Module 6 BDA

14 Trapezoidal rule N = 12 h = 0.25 No data x x/sqrt(x+1) In FreeMat: Result= ( ( ))= Lecture Module 6 BDA

15 Simpson 1/3 rule N = 12 h = 0.25 No data x x/sqrt(x+1) In FreeMat : Result= ( ( )+2( ))= Lecture Module 6 BDA

16 Simpson 3/8 rule N = 12 h = 0.25 No data x x/sqrt(x+1) In FreeMat : Result= (3) ( ( )+ 2( ))= Lecture Module 6 BDA

17 Example 6-2 A racing car velocity record from start to 12 seconds is shown below. You have to calculate the distance of the car from its start position in 12 seconds. The data is taken every 1 second. Use Simpson's rule 1/ speed (km/h) time (seconds) Lecture Module 6 BDA

18 N = 12 h= 1/3600 no t v(t) Result= 1 (170+ 4(640)+ 2(545))= km Lecture Module 6 BDA

19 Gauss Quadrature The main idea in Gauss Quadrature is to change the integration limits to natural (dimensionless) coordinate limits from -1 to 1 y y(x) ϕ x 1 ϕ(ξ) x n x Change to natural coordinates 1 1 Lecture Module 6 BDA ξ

20 Gauss Quadrature The main idea in Gauss Quadrature is to change the integration limits to natural (dimensionless) coordinate limits from -1 to 1 y y(x) x ξ = 1 2 [ (1 ξ) x 1 + (1+ ξ) x n ] ϕ(ξ)= y(x ξ ) ϕ x 1 ϕ(ξ) x n x I = x 1 n x 1 ϕ(ξ)d ξ Lecture Module 6 BDA ξ I = x n x 1 2 I ξ

21 1 I ξ = ϕ(ξ)d ξ 1 I ξ = R 1 ϕ(ξ 1 ) + R 2 ϕ(ξ 2 ) + + R n ϕ(ξ n ) ξ j R j is the location of the integration point j relative to the center is the weighting factor for point j relative to the center, and n is the number of points at which ϕ(ξ) is to be calculated Lecture Module 6 BDA

22 Gauss Quadrature Coefficients for Gaussian Quadrature n ±ξ j R j Lecture Module 6 BDA

23 Program to generate abscissa and weight for any number of integration points function [x,a] = GaussNodes(n,tol) % USAGE: [x,a] = GaussNodes(n,tol) % n = order of integration points % tol = error tolerance (default is 1.0e4*eps). format long; if nargin < 2; tol = 1.0e4*eps; end A = zeros(n,1); x = zeros(n,1); nroots = fix(n + 1)/2; for i = 1:nRoots t = cos(pi*(i )/(n + 0.5)); for j = i: 30 p0 = 1.0; p1 = t; for k = 1:n-1 p = ((2*k + 1)*t*p1 - k*p0)/(k + 1); p0 = p1;p1 = p; end dp = n *(p0 - t*p1)/(1 - t^2); dt = -p/dp; t = t + dt; if abs(dt) < tol x(i) = t; x(n-i+1) = -t; A(i) = 2/(1-(t^2))/(dp^2); A(n-i+1) = A(i); break end end if nroots == 1 x(i) =0; end end Lecture Module 6 BDA

24 Program to use Gauss Quadrature, any number of integration points function I = GaussQuadrature(func,a,b,n) % USAGE: I = gaussquad(func,a,b,n) % func = handle of function to be integrated. % for example --> ((sin(x)/x)^2) % a,b = integration limits. % n = order of integration points % I = integral result format long; c1 = (b + a)/2; c2 = (b - a)/2; [x,a] = GaussNodes(n); sum = 0; for i = 1:length(x) y = feval(func,c1 + c2*x(i)); sum = sum + A(i)*y; end I = c2*sum; Lecture Module 6 BDA

25 ϕ ϕ(ξ) Gauss Quadrature 1 point ξ weighting I ξ = R 1 ϕ(ξ 1 ) I ξ = 2.0 ϕ(0) Lecture Module 6 BDA

26 ϕ ϕ(ξ) Gauss Quadrature 2 points ξ weighting I ξ = R 1 ϕ(ξ 1 ) + R 2 ϕ(ξ 2 ) I ξ = 1.0 ϕ( ) ϕ( ) Lecture Module 6 BDA

27 ϕ ϕ(ξ) Gauss Quadrature 3 points ξ weighting I ξ = R 1 ϕ(ξ 1 ) + R 2 ϕ(ξ 2 ) + R 3 ϕ(ξ 3 ) I ξ = ϕ( ) ϕ(0.0) ϕ( ) Lecture Module 6 BDA

28 Example 6-3 Find the approximate value of 4 1 x x+ 1 dx By using : a) Gauss Quadrature 2 points b) Gauss Quadrature 3 points Lecture Module 6 BDA

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31 Example 6-4 Find the approximate value of x 2 x dx By using : a) Gauss Quadrature 4 points b) Check your result in Smath (Exact integration) c) Check your result, using Freemat (GaussQuadrature) Lecture Module 6 BDA

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34 Student Activity Lecture Module 6 BDA

35 Problem 6-A1 A military radar detected an unidentified flying object and found that the vertical velocity (m/s) can be expressed in a polynomial function, v(t)=20 t 3 +48t t If the flying object starts moving from the ground to the sky, calculate the distance of the flying object when it moves 10 seconds from the ground, by using (a) (b) 3-points Gauss quadrature. Simpson s rule with 10 divisions. (you have to select the appropriate Simpson's method) Lecture Module 6 BDA

36 Problem 6-A2 The velocity of a particle (m/s) at each 4 seconds time t, is measured and the recorded data is shown below Time velocity Time velocity a) Find the distance travel by the particle from 0 to 48 second by using trapezoidal rule b) Use Smath - Generate the polynomial function from 13 data then calculate the distance by Integrating it from 0 to 4 seconds Lecture Module 6 BDA

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