Simpson's Rule is a numerical method used to approximate the value of a definite integral. It works by dividing the interval of integration into smaller subintervals and using a quadratic polynomial to approximate the curve within each subinterval. The area under the curve is then approximated by summing the areas of these quadratic polynomials.
Simpson's Rule is most effective when the function being integrated is smooth and continuous. It is also more accurate than other numerical methods, such as the trapezoidal rule or midpoint rule, when the interval of integration is larger.
Simpson's Rule is a second-order method, meaning that it has an error of O(h^4) where h is the width of each subinterval. This means that the error decreases as h decreases, making Simpson's Rule more accurate than first-order methods like the trapezoidal rule.
No, Simpson's Rule can only be applied to definite integrals. It cannot be used for improper integrals or integrals with infinitely oscillating functions.
One limitation of Simpson's Rule is that it requires an even number of subintervals. It also may not work well for functions with sharp corners or discontinuities. In these cases, using a smaller h value or a different numerical method may be more effective.