The Trapezoidal Rule and Simpson’s Rule are both numerical methods used for approximating definite integrals. They offer a way to estimate the area under a curve when an analytical solution is not feasible or when dealing with functions that are too complex to integrate directly.

Trapezoidal Rule:

The Trapezoidal Rule approximates the area under a curve by dividing the interval into trapezoids and summing their areas. If we have a function  f(x) over the interval [a,b], the trapezoidal rule is given by:

∫ab​f(x)dx≈2h​[f(a)+2∑i=1n−1​f(xi​)+f(b)]

where:

• h=nb−a​ is the width of each subinterval,
• xi​=a+ih are the equally spaced points in the interval, and
• $n$ is the number of subdivisions.

Simpson’s Rule:

Simpson’s Rule approximates the function between each pair of points as a quadratic interpolant (a parabola) and integrates these parabolas to estimate the integral. For a function f(x) over the interval [a,b], Simpson’s rule is given by:

∫ab​f(x)dx≈3h​[f(a)+4∑i=1n/2−1​f(x2i​)+2∑i=1n/2​f(x2i−1​)+f(b)]

where:

• h=nb−a​ is the width of each subinterval,
• xi​=a+ih are the equally spaced points in the interval,
• $n$ is the number of subdivisions (must be even for Simpson’s rule).

Comparison:

• Accuracy: Simpson’s Rule generally provides a more accurate approximation compared to the Trapezoidal Rule, especially for functions with higher-order derivatives.
• Convergence: Simpson’s Rule typically converges to the exact integral more rapidly than the Trapezoidal Rule, meaning it requires fewer subdivisions for a given level of accuracy.
• Computational Complexity: Simpson’s Rule involves more complex calculations due to the quadratic interpolation, making it slightly more computationally expensive compared to the Trapezoidal Rule.

Both methods are widely used and serve as efficient tools for numerical integration, with the choice between them depending on the specific requirements of the problem at hand, such as accuracy and computational efficiency.