Simpson's Rule is a numerical method used to approximate the definite integral of a function. It is particularly useful when the exact integral is difficult or impossible to compute analytically. The method is based on the idea of approximating the integrand with a quadratic polynomial over small subintervals and is usually applied over a closed interval \([a, b]\).
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Simpson's rule integrates the quadratic interpolant through two endpoints and their midpoint. It is exact also on cubic polynomials. On , define the raw kernel , where is integral minus the rule. Then for and for . It is strictly negative inside the interval and integrates to . The Peano kernel theorem therefore gives the sharp bound ; attains equality. Scaling to endpoint spacing gives . The normalized Peano kernel is , so the external factorial must be omitted when using that normalization.