Simpson's 1/3 rule is a method for approximating the definite integral of a function. The formula is applied when you have an even number of intervals (which is essential) and it assumes that the curve between points is approximately a parabola. The rule works well for smooth and continuous functions.

The formula for Simpson's 1/3 rule to approximate a definite integral $\int_a^b f(x) dx$ is given by:

$I = \frac{h}{3} \left[ f(a) + 4 \sum_{i=1,3,5,\dots}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,\dots}^{n-2} f(x_i) + f(b) \right]$

Where:

• $h = \frac{b - a}{n}$ is the step size, and $n$ is the number of subintervals.
• $n$ must be even.

Here's a C program that implements Simpson's 1/3 rule for approximating a definite integral.

C Program: Simpson's 1/3 Rule

cCopy code
#include <stdio.h>
#include <math.h>

// Define the function to integrate
double f(double x) {
    // Example: f(x) = x^2 (You can modify this function as needed)
    return x * x;
}

// Function to implement Simpson's 1/3 Rule
double simpsons_one_third(double a, double b, int n) {
    // n must be even
    if (n % 2 != 0) {
        printf("n must be an even integer.\n");
        return -1;
    }
    
    double h = (b - a) / n;
    double sum = f(a) + f(b); // First and last terms

    // Applying Simpson's 1/3 rule
    for (int i = 1; i < n; i++) {
        double x = a + i * h;
        if (i % 2 == 0) {
            sum += 2 * f(x); // Even indexed terms
        } else {
            sum += 4 * f(x); // Odd indexed terms
        }
    }

    return (h / 3) * sum;
}

int main() {
    double a, b;
    int n;

    // Input the limits of integration a and b
    printf("Enter the lower limit a: ");
    scanf("%lf", &a);
    printf("Enter the upper limit b: ");
    scanf("%lf", &b);

    // Input the number of intervals (must be even)
    printf("Enter the number of intervals (n must be even): ");
    scanf("%d", &n);

    // Call Simpson's 1/3 rule function
    double result = simpsons_one_third(a, b, n);

    // Print the result
    if (result != -1) {
        printf("The approximate value of the integral is: %lf\n", result);
    }

    return 0;
}

Steps in the Code:

1. Define the function f(x): This is the mathematical function that you're integrating. In this case, it’s set as $f(x) = x^2$, but you can modify it for any function.
2. Simpson's 1/3 Rule implementation: The function simpsons_one_third(a, b, n) computes the approximation of the integral using Simpson's rule.
3. User input: The program asks for the lower limit a, the upper limit b, and the number of intervals n. The number of intervals n must be even for Simpson's rule to work properly.
4. Calculation and Output: The program calculates the approximate integral value and prints it.

Example Output:

Enter the lower limit a: 0
Enter the upper limit b: 2
Enter the number of intervals (n must be even): 4
The approximate value of the integral is: 2.666667

This example approximates the integral $\int_0^2 x^2 dx$, which has the exact value $\frac{8}{3} \approx 2.66667$.

Would you like further details or clarifications?
Here are some relative questions to deepen your understanding:

1. What is the mathematical background behind Simpson's 1/3 rule?
2. How can the function f(x) be changed for other types of integrals?
3. Why does Simpson’s 1/3 rule require an even number of intervals?
4. What is the difference between Simpson's rule and other numerical integration methods like the trapezoidal rule?
5. How does the accuracy of Simpson’s rule compare when increasing the number of intervals?

Tip: Increasing the number of intervals (n) can improve the accuracy of the approximation, but at a higher computational cost.