Question

Use Gaussian Quadrature to find the value of the integral of: f(x) = 79.13 / ( 5.30 + 2.24 * X * X) between X= -0.62 and X= 1.55 Integral using 2 terms Gaussian Quadrature is Integral using 3 terms Gaussian Quadrature is l__ Integral using 4 terms Gaussian Quadrature is Integral using 6 terms Gaussian Quadrature is Use the trapezoidal rule to find the value of the integral of: f(x) = 63.52 / ( 4.07 + 2.23 * X * x) between x= -0.15 and X= 1.56 Integral using the trapezoidal rule with 4 panels is Integral using the trapezoidal rule with 8 panels is panels is_ 6 Now, we define int (f,p) as the numerical integral of f(x) between X=-0.15 and X= 1.56 caculated by trapezoidal rule with p panels. Started from p[0]=4, a process is stopped by using the convergence criterion || [int(f,p[n])-int (,p[n-1])]/int (f,p[n]) ||<0.0000005, where p[n]=2*p [n-1], n=1,2,3 ...; Then the final result of int(E,p[n]) in this process is Use Simpson's rule to find the value of the integral of: f(x) = 59.18 / ( 8.79 + 1.24 * X* X) between x= -0.92 and x= 1.37 Integral using Simpson's rule with 4 panels is 8 Integral using Simpson's rule with 8 panels is_ _9 Now, we define int(1,p) as the numerical integral of f(x) between X=-0.92 and X= 1.37 caculated by Simpson's rule with p panels. Started from p[0]=4, a process is stopped by using the convergence criterion ||[int (E,p[n])-int (f,p[n-1])]/int (f,p[n]) ||<0.0000005, where p[n]=2*p [n-1], n=1,2,3 ... ;. Then the final result of int (E,p[n]) in this process is 10
Multiple choices

Answer 1

Answer Section Selection For Blank Number 1 + @ 26.7975 Selection For Blank Number 2 » @ 26.9188 Selection for Blomk Number 3 → 026.9263 Selection For Blank Humber 47 © 26.9242 Selection For Biomk Number 54 20.2891 selection For Blank Number 6 @ 20.3743 Selection For Blank Number 7 ② 20.4026 Selection for Blank Number 8 ® 14.4843 Selection For Blank Number og 14.4826 selection For Blank Number 1>@ 14.4825 .

%Matlab code for finding integration using Simpson Trapizoidal and Guassian Quadrature method clear all close all Function for which integration have to find func=@(x) 79.13./(5.30+2.24.*x.*x); fprintf('Function for which integration have to find f(x)=') disp(func) %Upper and lower limit of integration a=-0.62; b=1.55; fprintf('Upper limit a=ff and lower limit b= %f. \n', a,b) All n values n=[2 3 4 6); %Exact integral value ext_int=integral (func, a,b); fprintf('Exact integral value is %f.\n', ext_int) (func, a,b); def. In', ext_int) for i=1:length (n) Sintegration using Gaussian Quadrature method v_Gauss(i)=Gaussian_quadrature (func, a, b, n(i)); %Printing the result fprintf('For N=%d\n',n(i)) fprintf('\tIntegration using Gaussian quadrature method =% 2.15f\n \n',v_Gauss(i)) end Function for which integration have to find func=@(x) 63.52./(4.07+2.23.*x.*x); All n values n=[48]; fprintf('Function for which integration have to find f(x)=') disp(func) Upper and lower limit of integration a=-0.15; b=1.56; fprintf('Upper limit a=ff and lower limit b= %f.\n',a,b) for i=1:length(n) Sintegration using Gaussian Quadrature method v_trap (i)=trapizoidal(func, a,b,n(i)); %Printing the result fprintf('For N=%d\n',n(i)) fprintf('\tIntegration using Trapizoidal method =% 2.15f\n \n',v_trap (i))

end err=1; nn=4; while err>=0.0000005 v_trapl=trapizoidal(func, a, b, nn); v_trap2=trapizoidal (func, a, b, nn+1); err=abs( (v_trap2-v_trapl)/v_trap2); nn=nn+1; end fprintf('In Trapizoidal method Final result in this process is %f with n=%d\n',v_trap2, nn) Function for which integration have to find func=@(x) 59.18./(8.79+1.24.*x.*x); All n values n=[4 8 ]; fprintf('Function for which integration have to find f(x)=') disp(func) %Upper and lower limit of integration a=-0.92; b=1.37; fprintf('Upper limit a=%f and lower limit b= %.\n', a,b) for i=1:length(n) Sintegration using Gaussian Quadrature method v_simp (i)=simpson(func, a,b,n(i)); Printing the result fprintf('For N=%d\n',n(i)) fprintf('\tIntegration using Simpson method =% 2.15f\n \n',v_simp(i)) end err=1; nn=4; while err>=0.0000005 v_trapl=simpson (func, a,b, nn); v_trap2=simpson(func, a,b,nn+2); err=abs( (v_trap2-v_trapl)/v_trap2); nn=nn+1; end fprintf('In Trapizoidal method Final result in this process is %f with n=%d\n',v_trap2, nn) %%Matlab function for Trapizoidal integration function val=trapizoidal(func, a, b, N) % func is the function for integration % a is the lower limit of integration % b is the upper limit of integration % N number of rectangles to be used val=0; splits interval a to b into N+1 subintervals xx=linspace(a, b, N+1); dx=xx (2)-xx (1); %x interval

loop for Riemann integration for i=2: length(xx)-1 xxl=xx (i); val=val+dx*double(func(xx1)); end val=val+dx* (0.5*double(func(xx(1) ) ) +0.5*double(func(xx(end)))); end % % Matlab function for Simpson integration function val=simpson (func, a,b, N) % func is the function for integration % a is the lower limit of integration % b is the upper limit of integration % N number of rectangles to be used Esplits interval a to b into N+1 subintervals xx=linspace(a, b, N+1); dx=xx( 2 )-xx(1); %x interval val= (dx/3)* (double(func(xx(1)))+double(func( xx (end)))); %loop for Riemann integration for i=2: length(xx)-1 xxl=xx (i); if mod(i,2)==0 val=val+ (dx/3)*4*double(func (xxl)); else val=val+ (dx/3)*2*double(func (xxl)); end end end %Function for Gaussian quadrature function val=Gaussian_quadrature (func, a, b, N) olo % This script is for computing definite integrals using Legendre-Gauss % Quadrature. Computes the Legendre-Gauss nodes and weights on an interval % [a,b] with truncation order N % Suppose you have a continuous function f(x) which is defined on [a,b] % which you can evaluate at any xin [a,b]. Simply evaluate it at all of % the values contained in the x vector to obtain a vector f. Then compute % the definite integral using sum(f.*w); N=N-1; N1=N+1; N2=N+2; xu=linspace(-1,1,N1)'; % Initial guess

y=cos((2*(0:N)'+1) *pi/(2*N+2 ) ) +(0.27/N1)*sin(pi*xu*N/N2); % Legendre-Gauss Vandermonde Matrix L=zeros(N1,N2); Derivative of LGVM Lprzeros (N1, N2); % Compute the zeros of the N+1 Legendre Polynomial % using the recursion relation and the Newton-Raphson method y0=2; % Iterate until new points are uniformly within epsilon of old points while max( abs(y-yo))>eps L(:,1)=1; Lp (:,1)=0; L(:,2)=y; Lp (:,2)=1; for k=2:11 L(:, k+1)=( end (2*k-1) *y.*L(:,k)-(k-1)*L(:,k-1) )/k; Lp=(N2)*( L( :,N1)-y.*L(:,N2) )./(1-y.^2); yo=y; y=y0-L(:,N2)./Lp; end % Linear map from[-1,1] to [a,b] x=(a* (1-y) +b+ (1+y))/2; % Compute the weights w=(b-a)./((1-y.^2). *Lp.^2) * (N2/N1)^2; val=0; for ii=1:length(x) val=val+w(ii)* func(x(ii)); end end %%%%%%%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%%%%%% @(x) 79.13./ Function for which integration have to find f(x) = (5.30+2.24.*x.*x) Upper limit a=-0.620000 and lower limit b= 1.550000. Exact integral value is 26.924173. For N=2

Integration using Gaussian quadrature method =26.79 7483434492722 For N=3 Integration using Gaussian quadrature method =26.9187805462 73592 For N=4 Integration using Gaussian quadrature method =26.926294 7702 71483 For N=6 Integration using Gaussian quadrature method =26.924205955193827 @ (x) 63.52./ Function for which integration have to find f(x)= (4.07+2.23. *x. *x) Upper limit a=-0.150000 and lower limit b= 1.560000. For N=4 Integration using Trapizoidal method -20.289117379263775 For N=8 Integration using Trapizoidal method =20.374336 39 79 76432 In Trapizoidal method Final result in this process is 20.402217 with n=72 Function for which integration have to find f(x)= @(x)59.18./ (8.79+1.24. *x. *x) Upper limit a=-0.920000 and lower limit b= 1.370000. For N=4 Integration using Simpson method =14.484319254839484 For N=8 Integration using Simpson method =14.482584638085660 In Trapizoidal method Final result in this process is 14.482481 with n=15 Published with MATLAB® R2018a

%Matlab code for finding integration using Simpson Trapizoidal and Guassian
%Quadrature method
clear all
close all

%Function for which integration have to find
func=@(x) 79.13./(5.30+2.24.*x.*x);

fprintf('Function for which integration have to find f(x)=')
disp(func)
%Upper and lower limit of integration
a=-0.62; b=1.55;
fprintf('Upper limit a=%f and lower limit b= %f.\n',a,b)
%All n values
n=[2 3 4 6];

%Exact integral value
ext_int=integral(func,a,b);
fprintf('Exact integral value is %f.\n',ext_int)

for i=1:length(n)
  
    %integration using Gaussian Quadrature method
    v_Gauss(i)=Gaussian_quadrature(func,a,b,n(i));
    %Printing the result
    fprintf('For N=%d\n',n(i))
    fprintf('\tIntegration using Gaussian quadrature method =%2.15f\n\n',v_Gauss(i))
  
end

%Function for which integration have to find
func=@(x) 63.52./(4.07+2.23.*x.*x);
%All n values
n=[4 8];

fprintf('Function for which integration have to find f(x)=')
disp(func)
%Upper and lower limit of integration
a=-0.15; b=1.56;
fprintf('Upper limit a=%f and lower limit b= %f.\n',a,b)

for i=1:length(n)
  
    %integration using Gaussian Quadrature method
    v_trap(i)=trapizoidal(func,a,b,n(i));
    %Printing the result
    fprintf('For N=%d\n',n(i))
    fprintf('\tIntegration using Trapizoidal method =%2.15f\n\n',v_trap(i))
  
end

err=1; nn=4;
while err>=0.0000005
    v_trap1=trapizoidal(func,a,b,nn);
    v_trap2=trapizoidal(func,a,b,nn+1);
    err=abs((v_trap2-v_trap1)/v_trap2);
    nn=nn+1;
end
fprintf('In Trapizoidal method Final result in this process is %f with n=%d\n',v_trap2,nn)

%Function for which integration have to find
func=@(x) 59.18./(8.79+1.24.*x.*x);
%All n values
n=[4 8];

fprintf('Function for which integration have to find f(x)=')
disp(func)
%Upper and lower limit of integration
a=-0.92; b=1.37;
fprintf('Upper limit a=%f and lower limit b= %f.\n',a,b)

for i=1:length(n)
  
    %integration using Gaussian Quadrature method
    v_simp(i)=simpson(func,a,b,n(i));
    %Printing the result
    fprintf('For N=%d\n',n(i))
    fprintf('\tIntegration using Simpson method =%2.15f\n\n',v_simp(i))
  
end

err=1; nn=4;
while err>=0.0000005
    v_trap1=simpson(func,a,b,nn);
    v_trap2=simpson(func,a,b,nn+2);
    err=abs((v_trap2-v_trap1)/v_trap2);
    nn=nn+1;
end
fprintf('In Trapizoidal method Final result in this process is %f with n=%d\n',v_trap2,nn)

%%Matlab function for Trapizoidal integration
function val=trapizoidal(func,a,b,N)
    % func is the function for integration
    % a is the lower limit of integration
    % b is the upper limit of integration
    % N number of rectangles to be used
    val=0;
    %splits interval a to b into N+1 subintervals
    xx=linspace(a,b,N+1);
    dx=xx(2)-xx(1); %x interval
    %loop for Riemann integration
        for i=2:length(xx)-1
            xx1=xx(i);
            val=val+dx*double(func(xx1));
        end
       val=val+dx*(0.5*double(func(xx(1)))+0.5*double(func(xx(end))));
end

%%Matlab function for Simpson integration
function val=simpson(func,a,b,N)
    % func is the function for integration
    % a is the lower limit of integration
    % b is the upper limit of integration
    % N number of rectangles to be used
    %splits interval a to b into N+1 subintervals
    xx=linspace(a,b,N+1);
    dx=xx(2)-xx(1); %x interval
    val=(dx/3)*(double(func(xx(1)))+double(func(xx(end))));
    %loop for Riemann integration
        for i=2:length(xx)-1
            xx1=xx(i);
            if mod(i,2)==0
                val=val+(dx/3)*4*double(func(xx1));
            else
                val=val+(dx/3)*2*double(func(xx1));
              
            end
        end    
end

%Function for Gaussian quadrature
function val=Gaussian_quadrature(func,a,b,N)

%
% This script is for computing definite integrals using Legendre-Gauss
% Quadrature. Computes the Legendre-Gauss nodes and weights on an interval
% [a,b] with truncation order N
%
% Suppose you have a continuous function f(x) which is defined on [a,b]
% which you can evaluate at any x in [a,b]. Simply evaluate it at all of
% the values contained in the x vector to obtain a vector f. Then compute
% the definite integral using sum(f.*w);
%
    N=N-1;
    N1=N+1; N2=N+2;

    xu=linspace(-1,1,N1)';

    % Initial guess
    y=cos((2*(0:N)'+1)*pi/(2*N+2))+(0.27/N1)*sin(pi*xu*N/N2);

    % Legendre-Gauss Vandermonde Matrix
    L=zeros(N1,N2);

    % Derivative of LGVM
    Lp=zeros(N1,N2);

    % Compute the zeros of the N+1 Legendre Polynomial
    % using the recursion relation and the Newton-Raphson method

    y0=2;

    % Iterate until new points are uniformly within epsilon of old points
    while max(abs(y-y0))>eps

        L(:,1)=1;
        Lp(:,1)=0;

        L(:,2)=y;
        Lp(:,2)=1;

        for k=2:N1
            L(:,k+1)=( (2*k-1)*y.*L(:,k)-(k-1)*L(:,k-1) )/k;
        end

        Lp=(N2)*( L(:,N1)-y.*L(:,N2) )./(1-y.^2);

        y0=y;
        y=y0-L(:,N2)./Lp;

    end

    % Linear map from[-1,1] to [a,b]
    x=(a*(1-y)+b*(1+y))/2;    

    % Compute the weights
    w=(b-a)./((1-y.^2).*Lp.^2)*(N2/N1)^2;
  
    val=0;
    for ii=1:length(x)
        val=val+w(ii)*func(x(ii));
    end
end
%%%%%%%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%%%%%%