如何在Python中无导入实现梯形积分并编写指定参数的积分函数？

Let's fill in the core logic for the integral function to compute the trapezoidal estimate of a function's integral over a given interval. Here's the step-by-step solution:

Core Logic Breakdown #

The trapezoidal rule splits the interval (x0, xn) into n equal-width subintervals. Each subinterval forms a trapezoid, and the total integral is the sum of all these trapezoid areas. The formula we’ll use is:

Integral ≈ (h/2) * [f(x0) + 2f(x1) + 2f(x2) + ... + 2*f(xn-1) + f(xn)]
where h = (xn - x0)/n is the width of each subinterval.

Complete Code with Integrated Core Logic #

def f1(x): return x * 4 * (1 - x) * 4 / (1 + x ** 2)
def f2(x): 
    e = 2.718281828459045 
    return e ** -x
def integral(f, interval, n):
    ''' computes estimate of integral of f over interval based on deviding interval into n trapeziums '''
    if not (callable(f) and isinstance(interval, tuple) and isinstance(n, int)):
        return
    if not (len(interval)==2 and isinstance(interval[0], float) and isinstance(interval[1], float) and interval[0]<interval[1]):
        return
    
    # Core trapezoidal integration logic starts here
    x0, xn = interval
    h = (xn - x0) / n  # Calculate width of each trapezoid
    total = f(x0) + f(xn)  # Start with first and last function values (only counted once)
    
    # Add 2*f(x) for all intermediate points (shared between two trapezoids)
    for i in range(1, n):
        current_x = x0 + i * h
        total += 2 * f(current_x)
    
    # Compute final integral estimate
    integral_estimate = total * h / 2
    # Core logic ends here
    
    return integral_estimate

# Test the example case
print(integral(lambda x : x, (1.0, 2.0), 100))  # Outputs ~1.5, matching the exact integral result

Key Details #

• We first extract the interval bounds and calculate the subinterval width h.
• The sum initializes with the first and last function values because each only appears in one trapezoid.
• For every intermediate point, we add twice the function value—since these points are the "shared edge" between two adjacent trapezoids.
• Finally, multiply the total sum by h/2 to get the final integral estimate.

内容的提问来源于stack exchange，提问作者orsa