Simpson’s method is a more accurate numerical integration technique. It approximates the function using parabolas on each subinterval. For the method to work, the number of subintervals must be even: \( n = 2m \).

The formula for Simpson’s 1/3 rule (composite version) is:

I ≈ (h / 3) × [f(a) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + 4f(xₙ₋₁) + f(b)]

Simpson’s method combines the strengths of the rectangle and trapezoid methods. Its error is of fourth order, making it very efficient for smooth functions.

Remarks:

• All odd-indexed terms (x₁, x₃, …, xₙ₋₁) are multiplied by 4, while even-indexed terms (x₂, x₄, …, xₙ₋₂) are multiplied by 2.
• The first and last function values (f(a) and f(b)) appear only once.
• Simpson’s method is exact for all polynomials of degree ≤ 3.