1/3 simpson rule for an integration in php

To implement the 1/3 Simpson's rule for numerical integration in PHP, you can follow these steps:

1. Define the function you want to integrate, let's call it f(x).
2. Specify the interval limits of integration, let's say a and b.
3. Determine the number of subintervals, n, which should be an even number.
4. Compute the width of each subinterval, h, using the formula h = (b - a) / n.
5. Initialize the variables sum and x to zero.
6. Iterate i from 0 to n, with step size 2.
7. Inside the loop, calculate x as a + i * h.
8. Compute the corresponding y value for x using the function f(x).
9. Add the calculated value of y to sum.
10. Repeat steps 6-9 but with i starting from 1 and ending at n-1, with step size 2.
11. Compute the final result using the formula result = (h / 3) * (f(a) + f(b) + 4 * sum).
12. Return the calculated result.

Here is the PHP code implementing the 1/3 Simpson's rule:

main.php
function f($x) {
    // Define your function here
    return // calculation of f(x) using the given expression
}

function integrate($a, $b, $n) {
    $h = ($b - $a) / $n;
    $sum = 0;
    
    for ($i = 0; $i <= $n; $i += 2) {
        $x = $a + $i * $h;
        $sum += f($x);
    }

    for ($i = 1; $i < $n; $i += 2) {
        $x = $a + $i * $h;
        $sum += 4 * f($x);
    }

    $result = ($h / 3) * (f($a) + f($b) + 2 * $sum);
    return $result;
}

// Example usage
$a = 0; // lower limit
$b = 2; // upper limit
$n = 100; // number of subintervals

$result = integrate($a, $b, $n);
echo "Result: " . $result;
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Make sure to replace f($x) with your own function to integrate, and adjust the values of $a, $b, and $n according to your specific integration problem.

Note that this implementation assumes that the function is continuous and can be evaluated at any given point within the specified interval.