Integration over an infinite interval. Integrate exp(−x) from x = 0 to x = ∞, by numerically integrating from x = 0 to increasing values of x. Plot a graph of the error vs the value of x you integrate to. Compare results for a fixed step size and a fixed number of steps. use matlab to finish show me the codes

Here is a simple MATLAB code to perform the task:

% Define the function
f = @(x) exp(-x);

% Define the exact value of the integral
exact = 1;

% Define the range of x values
x_max = 1:1:100;

% Initialize arrays to store the errors
error_fixed_step = zeros(size(x_max));
error_fixed_num_steps = zeros(size(x_max));

% Define the fixed step size and number of steps
fixed_step_size = 0.01;
fixed_num_steps = 1000;

% Loop over the x values
for i = 1:length(x_max)
    % Calculate the integral with a fixed step size
    x = 0:fixed_step_size:x_max(i);
    y = f(x);
    integral_fixed_step = trapz(x, y);
    error_fixed_step(i) = abs(integral_fixed_step - exact);
    
    % Calculate the integral with a fixed number of steps
    x = linspace(0, x_max(i), fixed_num_steps);
    y = f(x);
    integral_fixed_num_steps = trapz(x, y);
    error_fixed_num_steps(i) = abs(integral_fixed_num_steps - exact);
end

% Plot the errors
figure;
plot(x_max, error_fixed_step, 'r', 'LineWidth', 2);
hold on;
plot(x_max, error_fixed_num_steps, 'b', 'LineWidth', 2);
xlabel('x');
ylabel('Error');
legend('Fixed step size', 'Fixed number of steps');
title('Error in numerical integration of exp(-x) over increasing intervals');
grid on;

This code first defines the function to be integrated and the exact value of the integral. It then initializes arrays to store the errors for the two methods. It loops over the x values, calculating the integral and the error for each method at each x value. Finally, it plots the errors as a function of x. The red line represents the error for the fixed step size method, and the blue line represents the error for the fixed number of steps method.