Integration over discontinuities. Integrate the function f(x) = 0 for x < 1, f(x) = exp(x − 1) + exp(2 − x) for 1 ≤ x ≤ 2, f(x) = 0 for x > 2 from x = 0 to x = 3, for varying step sizes, using the trapezoidal method. Plot the error as a function of step size. A Matlab function calculating this function, f_r13a.m, is available on Blackboard. Note that the integral is equal to 2e − 1. use matlab and show me the codes

Here is a simple MATLAB code that calculates the integral using the trapezoidal method and plots the error as a function of step size.

% Define the function
f = @(x) (x >= 1 & x <= 2) .* (exp(x - 1) + exp(2 - x));

% Exact integral value
exact_integral = 2*exp(1) - 1;

% Initialize arrays to store step sizes and errors
step_sizes = logspace(-2, 0, 50);
errors = zeros(size(step_sizes));

% Loop over different step sizes
for i = 1:length(step_sizes)
    % Current step size
    h = step_sizes(i);
    
    % Define the x values
    x = 0:h:3;
    
    % Calculate the integral using the trapezoidal method
    integral = h * (0.5*f(x(1)) + sum(f(x(2:end-1))) + 0.5*f(x(end)));
    
    % Calculate the error
    errors(i) = abs(integral - exact_integral);
end

% Plot the error as a function of step size
loglog(step_sizes, errors, 'b.-');
xlabel('Step size');
ylabel('Error');
title('Error as a function of step size using the trapezoidal method');
grid on;

This code first defines the function f using an anonymous function. It then calculates the integral for different step sizes using the trapezoidal method and stores the errors in an array. Finally, it plots the error as a function of step size on a log-log plot.

Please note that you need to adjust the step sizes and the range of integration according to your specific problem. Also, the function f is defined using logical indexing, which might not be familiar to beginners.