Simpson's Rule

• Simpson’s Rule is a technique utilized for numerical integration. It gives an estimate for the value of a definite integral.

• It is named after its inventor, Thomas Simpson, who was an 18th Century British mathematician.

• The rule provides an approximation of the definite integral of a function, which is more accurate than the approximation produced using the Trapezium Rule.

• When applying Simpson’s Rule, the interval over which we integrate (b - a) is split into even sub-intervals.

• The approximation generated by Simpson’s Rule is attained by calculating the area of a series of parabolas which straddle these sub-intervals.

• Unlike the Trapezium Rule, which utilises straight line segments to approximate the function, Simpson’s Rule uses quadratic functions, making it generally more accurate.

• The basic formula for Simpson’s Rule is given as: ∫ (from a to b) f(x) dx ≈ (b - a)/6[n* (f(a) + 4f((a+b)/2) + f(b)]

• Notice that the formula for Simpson’s Rule includes an f((a+b)/2) term, signifying that Simpson’s Rule takes into account the midpoint of the interval, in addition to the end points.

• In order to use Simpson’s Rule, one needs to have a function which is integrable and has a continuous second derivative.

• The more segments you use (i.e., the smaller the width of each segment), the more accurate your result will generally be. However, there is a trade-off in terms of computational resources, as using more segments requires more computation.

• It’s essential to remember that Simpson’s Rule is just an approximation. The real value of the definite integral could be slightly more or less than the value calculated using the rule.

• Simpson’s Rule is generally more accurate than using the Midpoint or Trapezium Rule when the being-integrated function is well-approximated by a series of parabolas.

• The effectiveness of Simpson’s Rule is significantly reduced if the function has an inflection point within the interval [a,b].