Numerical Integration: Simpson and Trapezoid

Integration is a fundamental concept of calculus that calculates the accumulated value of a function over a given interval. Formally, the definite integral of a continuous function \( f(x) \) over the interval \([a, b]\) is expressed as:

This operation measures the net area under the curve of the function \( f(x) \), considering areas above the x-axis as positive and those below as negative. When the function does not have a simple analytical form or the integral cannot be directly solved, numerical methods like the Trapezoid and Simpson methods are used to approximate this area using sums of simple geometric shapes.