Compute the definite integral shown below using the Rectangl

Compute the definite integral shown below using the Rectangle Method and the Trapeziodal Method. Compare your approximate answers to the exact solution (-2 pi). Calculate the relative error in both approximations, and plot the relative error as a function of N, the number of subintervals in the domain. Use a logarithmic scale for the y axis. Plot from N=5 to N=500. Fully annotate the plot. I = integral^2 pi_0 x sin(x) dx

Solution

Following is the required matlab code:

clc; clear;

%rectangle method
a = 0;
b = 2*pi;
Error = zeros( 500, 1 );
Ns = zeros( 500, 1 );

for N=5:500
    h = (b - a)/N;
    answer = 0;
    xn = a;
    for i=0:N-1
        %find value of x*sin(x) for xn
        answer = answer + xn*sin(xn);
        xn = xn + h;
    end
    answer = answer*h;
  
    relativeError = abs((-2*pi) - answer)/abs(-2*pi);

    Error(N, 1) = log(relativeError); %using logarithmic scale
    Ns( N, 1 ) = N;
end
figure;
plot( Ns(5:end) , Error(5:end) );
title(\'log(Relative Error), using rectangle method vs N\');
xlabel(\'N\');
ylabel(\'log(Relative Error)\');

%trapezoidal method
a = 0;
b = 2*pi;
Error = zeros( 500, 1 );
Ns = zeros( 500, 1 );

for N=5:500
    h = (b - a)/N;
    answer = 0;
    xk = a;
    xkplus1 = a + h;
    for i=0:N-1
        answer = answer + ( xk*sin(xk) + xkplus1*sin(xkplus1) );
        xk = xkplus1;
        xkplus1 = xkplus1 + h;
    end
    answer = answer*(h/2);
  
    relativeError = abs((-2*pi) - answer)/abs(-2*pi);

    Error(N, 1) = log(relativeError); %using logarithmic scale
    Ns( N, 1 ) = N;
end
figure;
plot( Ns(5:end) , Error(5:end) );
title(\'log(Relative Error), using trapezoidal method vs N\');
xlabel(\'N\');
ylabel(\'log(Relative Error)\');