Numerical Integration

Quadrature and Basic Rules

Many definite integrals lack an antiderivative, like the Gaussian $\int e^{-x^2} dx$. When the Fundamental Theorem fails, numerical integration (quadrature) approximates the area using function values.

Simpson's rule fits a parabola through each point triple. Because parabolas hug curves tightly, halving $h$ cuts its error by sixteen. It is exact for polynomials up to degree three.

Accuracy Limits and Pitfalls

The golden principle: finer subdivisions ($h$) and higher-order rules drive faster convergence.

Common Pitfall: Assuming more subdivisions always improve accuracy. Shrinking $h$ reduces truncation error, but summing millions of tiny terms allows round-off error to accumulate and eventually dominate the final result.

The Solution: Do not blindly shrink $h$ into the round-off zone. Instead, use a higher-order method like Simpson's rule to achieve high accuracy at a modest step size.