simpson's 1/3 rule in matlab

To implement Simpson's 1/3 rule in MATLAB, you can follow these steps:

1. Define the function f(x) that you want to integrate.
2. Define the limits of integration, that is, the lower limit a and the upper limit b.
3. Choose the number of intervals n for the approximation. Note that n should be an even number.
4. Calculate the step size h using the formula h = (b - a) / n.
5. Initialize the variable sum to 0, which will store the sum of the function values.
6. Use a loop to compute the sum of the function values at each interval. Start the loop at i = 1 and increment i by 1 until i < n.
7. For each iteration of the loop, calculate the x-coordinate of the current subinterval midpoint using the formula x = a + (i-0.5) * h.
8. Add the corresponding function value f(x) to the sum variable. The value of f(x) will depend on your specific function f(x).
9. After the loop, calculate the approximate integral using the formula integral = h / 3 * (f(a) + 4 * sum + f(b)).
10. Finally, display or use the value of integral as required.

Here is a MATLAB code snippet that demonstrates the implementation:

main.m
% Define the function f(x)
f = @(x) sin(x);

% Define the limits of integration
a = 0;
b = pi;

% Choose the number of intervals
n = 100; % Should be an even number

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

% Initialize the sum variable
sum = 0;

% Loop to compute the sum of function values
for i = 1:n
    x = a + (i - 0.5) * h;
    sum = sum + f(x);
end

% Calculate the approximate integral
integral = h / 3 * (f(a) + 4 * sum + f(b));

% Display the integral
disp(integral);
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You can replace the function f(x) with your own function that you want to integrate. The code snippet above demonstrates the integration of the sin(x) function from 0 to π using 100 intervals.