4.7. Python Code Example (Trapezoidal Rule)

def f(x):
    return x**2 # Example function

In this chunk, we define a function f(x) that takes a variable x as input and returns the square of x, which is x**2. This function serves as our example function to integrate using the Trapezoidal Rule. The comment # Example function indicates that this is just a demonstration of how we can define a function in Python.

Think of defining a function like creating a recipe for a dish. Just as you write down the steps to prepare the meal, in programming, you set out a sequence of operations that the computer will perform when you call the function.

def trapezoidal(a, b, n):
    h = (b - a) / n
    result = f(a) + f(b)
    for i in range(1, n):
        result += 2 * f(a + i * h)
    return (h / 2) * result

Here, we implement the Trapezoidal Rule in a function called trapezoidal, which takes three parameters: a (the start of the interval), b (the end of the interval), and n (the number of sub-intervals). We first calculate the width of each trapezoid, h, by dividing the total length of the interval (b - a) by the number of sub-intervals n. We initialize result with the sum of the function values at the endpoints f(a) and f(b). Then, we loop through the intermediate points, adding twice the function value at each of these points to result, as each is counted in two trapezoids. Finally, we multiply the total result by h / 2 to get the final integration result.

Imagine you are piecing together a jigsaw puzzle. The edges are the corners of your interval (like f(a) and f(b)), while the pieces in between represent the intermediate points. Just as you would need to carefully connect the pieces to see the whole picture, here, we are carefully summing up the trapezoids to approximate the area under the curve.

# Example usage
print(trapezoidal(0, 1, 10)) # Approximate ∫0^1 x² dx

This chunk demonstrates how to use the trapezoidal function we defined earlier. By calling print(trapezoidal(0, 1, 10)), we are asking Python to calculate the approximate integral of the function f(x) = x² from 0 to 1 using 10 sub-intervals. The output of this function will give us an approximation of the definite integral, which is a numerical representation of the area under the curve of the function in that interval.

If we think of the integration as measuring the area of a field, using the trapezoidal rule is like sending people to measure different sections and then summing up those measurements. The more sections (sub-intervals) you measure, the more accurate your total area estimate will be.

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