The Newton-Cotes formulas, the most commonly used numerical integration methods, approximate the integration of a complicated function by replacing the function with many polynomials across the integration interval. The integration of the original function can then be obtained by summing up all polynomials whose "areas" are calculated by the weighting coefficients and the values of the function at the nodal points.

Trapezoidal Rule:

Simpson's Rule:

Simpson's 3/8 Rule:

Boole's Rule: