Question

Consider the following function sin(x) Using the following parameters in your functions: -func: the function/equation that you are required to integrate -a, b: the integration limits n: the number of points to be used for the integration I:Integral estimate a) Write a function capable of performing numerical integration of h(x) using the composite trapezoidal rule. Use your function to integration the equation with 9 points. Write a function capable of performing numerical integration of h(x) using the composite Simpson's 1/3 rule. Use your function to integrate the equation with 9 points. Use quad command to integrate the equation. Calculate the percentage error of the solutions from part (a) and (b) compared to the quad solution. b) c) Task 2 Write an m-file capable of performing numerical integration for the equation f(x)(6+3 cos(x)) dx 0 using the functions written in Task 1 (Trapezoidal and Simpson's 1/3). The program is interactive and should include the following: a) Ask the user to input which rule to use (e.g. 1' for composite trapezoidal and 2 b)Ask the user to input the number of points for which the integration is calculated c) Print the percentage error between the approximated solution and the solution for composite Simpson's 1/3 rule) The program should notify the user if an invalid number of points are used obtained using the quad function.

Use Matlab code

Answer 1

function I = trapezoidalRule(func,a,b,n)

h = (b-a)/n; % step size
F0 = func(a); Fn = func(b); %first and last elements
x = a;
sigma = 0;
for k=1:n-1
x = x+h;
sigma = sigma + func(x); % sum of intermediate elements
end

I = (h/2)*(F0+ 2*sigma + Fn); %final integral approximation

end

---------------------------------------------------------

function I = simpsonsRule(func,a,b,n)

h = (b-a)/n; %step size

F = 0;
for i=0:n
if i==0 || i==n % first and last coefficients
c =1;
elseif mod(i,2)~=0 %odd coefficients
c = 4;
elseif mod(i,2)==0 %even coefficients
c = 2;
end
  
F = F + c*func(a+i*h); % computing the sum
end

I = (h/3)*F; % final integral approximation
end

---------------------------------------------------------------

%driver file

clear
clc

% Task 1
func = @(x) 3 + (x.*sin(x))./(x-1);
a = 2; b = 7; n = 9;

% using trapezoidal rule
Itrap = trapezoidalRule(func,a,b,n)

% using simpson's rule
Isimp = simpsonsRule(func,a,b,n)

% using quad function
Iquad = quad(func,a,b)

%percentage error
ItrapPercentError = (abs(Iquad-Itrap)/Iquad)*100
IsimpPercentError = (abs(Iquad-Isimp)/Iquad)*100

%--------------------------------------------------------------

% Task 2

func = @(x) 6 + 3.*cos(x);
a = 0; b = pi/2;

disp('1 - composite trapezoidal rule')
disp('2 - composite simpson''s 1/3 rule')
rule = input('Which rule to use? ');
n = input('Enter number of points n: ');

if n>=2
switch rule
case 1
Itrap = trapezoidalRule(func,a,b,n)
Iquad = quad(func,a,b)
%percentage error
ItrapPercentError = (abs(Iquad-Itrap)/Iquad)*100
case 2
Isimp = simpsonsRule(func,a,b,n)
Iquad = quad(func,a,b)
IsimpPercentError = (abs(Iquad-Isimp)/Iquad)*100
otherwise
disp('invalid input')
end

else
disp('invalid number of points')
end

Itrap 13.8139 Isimp 13.0962 Iquad 13.7474 ItrapPercentError - 0.4841 IsimpPercentError 4.7369

1-composite trapezoidal rule 2 composite simpson's 1/3 rule Which rule to use? 1 Enter number of points n: 20 Itrap 12.4232 |quad 12.4248 ItrapPercentError 0.0124

1 composite trapezoidal rule 2 composite simpson's 1/3 rule Which rule to use? 2 Enter number of points n: 20 Isimp = 12.4248 Iquad - 12.4248 İsimpPercentError 5.1592e-06

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