That's a solution thoughDick said:
Simpson 1/8 Integration is a numerical method used for approximating definite integrals. It is based on dividing the interval of integration into smaller subintervals and using a quadratic polynomial to approximate the curve within each subinterval. This method is more accurate than the trapezoidal rule and is commonly used in scientific and engineering applications.
To use Simpson 1/8 Integration, the interval of integration is divided into an even number of subintervals. Then, a quadratic polynomial is fit to each pair of adjacent subintervals. The integral of the polynomial within each pair of subintervals is then calculated and summed to approximate the overall integral of the function over the entire interval. The accuracy of the approximation increases as the number of subintervals increases.
Simpson 1/8 Integration is more accurate than the trapezoidal rule and requires fewer subintervals to achieve the same level of accuracy. It also has a faster convergence rate, meaning that the approximation approaches the true value of the integral more quickly. Additionally, it can be used for functions with more complex shapes and does not require the function to be continuous.
Simpson 1/8 Integration can only be used for definite integrals and requires the function to have a continuous second derivative. It also does not perform well for functions with sharp corners or discontinuities. In these cases, a different numerical method may be more appropriate.
The error of Simpson 1/8 Integration can be calculated using the error formula: E = (b-a)(h^5/90)*f^(4)(c), where a and b are the limits of integration, h is the width of each subinterval, and f^(4)(c) is the fourth derivative of the function evaluated at some point c within the interval of integration. The error decreases as the number of subintervals increases, and the approximation approaches the true value of the integral.