The trapezoid rule and Simpson's rule are numerical methods for approximating definite integrals by partitioning the interval of integration into n equal subintervals and combining function values at the partition points with fixed weighting coefficients. The trapezoid rule approximates the region under the curve as a series of trapezoids (endpoints weighted by 1/2, interior points weighted by 1), while Simpson's rule uses a weighted combination with alternating coefficients (1, 4, 2, 4, ..., 2, 4, 1, scaled by Δx/3) and requires an even number of subintervals. This belongs to numerical integration theory within calculus, providing approximation techniques for integrals that are difficult or impossible to evaluate via closed-form antiderivatives.
The trapezoid rule and Simpson's rule are numerical methods for approximating definite integrals by partitioning the interval of integration into n equal subintervals and combining function values at…