Python Integral Calculator Without Scipy

What is an Integral?

An integral represents the area under a curve between two points. In calculus, integrals are used to find accumulations such as area, volume, displacement, and more. There are two main types of integrals:

• Definite Integral: Calculates the exact area under a curve between two points.
• Indefinite Integral: Represents the antiderivative of a function, which is the family of all functions whose derivative is the original function.

For numerical computation, we often use approximation methods when exact solutions are difficult to find.

Numerical Integration Methods

When an exact solution isn't available, numerical methods approximate the integral. Common methods include:

1. Trapezoidal Rule: Approximates the area under the curve using trapezoids.
2. Simpson's Rule: Uses parabolas to approximate the area, generally more accurate than the trapezoidal rule.
3. Rectangle Method: Uses rectangles to approximate the area (left, right, or midpoint).

Python Implementation Without SciPy

Here's how to implement numerical integration in Python without external libraries:

def trapezoidal_rule(f, a, b, n=1000):
    h = (b - a) / n
    integral = (f(a) + f(b)) / 2
    for i in range(1, n):
        integral += f(a + i * h)
    return integral * h

def simpsons_rule(f, a, b, n=1000):
    if n % 2 != 0:
        n += 1  # Simpson's rule requires even number of intervals
    h = (b - a) / n
    integral = f(a) + f(b)
    for i in range(1, n):
        x = a + i * h
        if i % 2 == 0:
            integral += 2 * f(x)
        else:
            integral += 4 * f(x)
    return integral * h / 3

Example Calculation

Let's calculate the integral of f(x) = x² from 0 to 1 using both methods:

The exact value of ∫₀¹ x² dx is 1/3 ≈ 0.3333. Both methods provide good approximations with sufficient intervals.