Simpson’s Rule

Simpson’s Rule is a numerical integration technique used to approximate the definite integral of a function. It is based on the idea of approximating the function by a series of parabolas, providing a more accurate estimate of the integral than using simple rectangles or trapezoids, as in the Trapezoidal Rule.

To apply Simpson’s Rule, the interval of integration is divided into an even number of equally spaced subintervals. Then, a quadratic polynomial is fit to each pair of adjacent subintervals, and the area under each parabola is calculated. The sum of these areas provides an approximation of the definite integral. Simpson’s Rule is particularly effective for functions that are relatively smooth and have continuous second derivatives over the interval of integration.

While Simpson’s Rule provides a more accurate approximation than the Trapezoidal Rule, it is still subject to errors. The accuracy can be improved by increasing the number of subintervals or using other, more advanced numerical integration techniques, such as Gaussian quadrature or adaptive integration methods.