Editing Equations, Formulas and Algorithms (section)

==== Numerical Integration ====
Numerical integration methods approximate the integral of a function over a specified interval. Common numerical integration techniques include the trapezoidal rule, Simpson's rule, and Gaussian quadrature. These methods are used to compute integrals in situations where analytical solutions are not feasible.

'''Example: Trapezoidal Rule'''
The trapezoidal rule approximates the integral of a function <math> f(x) </math> over the interval <math>[a, b]</math> as follows:
<math mode="display"> \int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left( f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) \right) </math>
where <math> h = \frac{b - a}{n} </math> is the width of each subinterval and <math> x_i = a + ih </math> are the points at which the function is evaluated.

'''Source Code Example (Python):'''
<syntaxhighlight lang="python">
def trapezoidal_rule(f, a, b, n):
    h = (b - a) / n
    sum_val = 0.5 * (f(a) + f(b))
    for i in range(1, n):
        sum_val += f(a + i * h)
    return h * sum_val

# Example usage:
def func(x):
    return x**2

result = trapezoidal_rule(func, 0, 1, 100)
print("Approximated integral using Trapezoidal Rule:", result)
</syntaxhighlight>