Numerical Integration - Simpson’s 1/3rd Rule

Program 1

Simpson’s 1/3rd Rule

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Mathematical Problem Formulation

Integration by Simpson method

Program 2

Simpson’s 1/3rd Rule

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Mathematical Problem Formulation

Integration by Simpson method

Program 3

Simpson’s 1/3rd Rule

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import numpy as np

def simpsons_1_3_rule(f, a, b, n):
    """
    Approximate the integral of a function using Simpson's 1/3 rule.

Parameters:
    f : function
        The function to integrate.
    a : float
        The lower bound of the integration interval.
    b : float
        The upper bound of the integration interval.
    n : int
        The number of subintervals to use (must be even).

Returns:
    float
        The approximate value of the integral.
    """
    # Ensure the number of subintervals is even
    if n % 2 != 0:
        raise ValueError("Number of subintervals must be even.")
    
    # Calculate the width of each subinterval
    h = (b - a) / n
    
    # Evaluate the function at the endpoints
    integral = f(a) + f(b)
    
    # Sum the function values at the odd and even indices
    for i in range(1, n, 2):
        integral += 4 * f(a + i * h)
    for i in range(2, n, 2):
        integral += 2 * f(a + i * h)
    
    # Multiply by the width of each subinterval and divide by 3
    integral *= h / 3.0
    
    return integral

def f(x):
    return np.sin(x)  # Example function: sine of x
    
    # Define the integration bounds and number of subintervals
a = 0          # Lower bound
b = np.pi      # Upper bound
n = 10         # Number of subintervals (must be even)
    
    # Calculate the integral
result = simpsons_1_3_rule(f, a, b, n)
print(f"Approximate integral: {result:.6f}")

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Integration by Simpson method

Ch. 11.2 Simpson’s 1/3rd Rule

Simpson’s 1/3rd Rule

Mathematical Problem Formulation

Simpson’s 1/3rd Rule

Mathematical Problem Formulation

Simpson’s 1/3rd Rule

Mathematical Problem Formulation