Simpson's Rule

Understanding Simpson’s Rule

• Simpson’s Rule is a method of numerical integration which is an approximation technique to find definite integrals and areas under curves.
• It is an application of the Newton-Cotes method of numerical integration.
• The rule offers a way to improve the approximation by taking parabolic arcs between groups of three points, rather than straight lines between two points.
• Simpson’s Rule is more accurate than the Midpoint Rule or Trapezoidal Rule, which are other numerical methods of integration.

Fundamental Formula of Simpson’s Rule

• The formula for Simpson’s Rule is: I = h/3 [y0 + 4(y1 + y3 + y5 + … + yn-1) + 2(y2 + y4 + y6 + … + yn-2) + yn]
• In this equation, I is the approximation of the integral, h is the width of each strip into which the area under the curve is divided (also called step size), n is the total number of strips (which should be an even number for Simpson’s Rule to be applied), and y0, y1, …, yn are the y-values of the function at the points determined by the step size.
• Note that the first and last y-values are included only once in the formula, whereas the middle y-values are either multiplied by 4 (if odd) or by 2 (if even).

Applying Simpson’s Rule

• To apply Simpson’s Rule, you first need to define the interval of integration and the number of strips to use.
• Calculate h by subtracting the lower limit from the upper limit and dividing by the number of strips.
• Determine the y-values based on the function you’re integrating for each point as per the width h.
• Substitute these values into the Simpson’s Rule formula.
• Carry out any multiplication and addition to find your numerical integration approximation.

Limitations of Simpson’s Rule

• Simpson’s Rule provides an approximation, not the exact integral solution.
• Accuracy of Simpson’s Rule depends on the function being integrated and how closely it can be approximated by parabolas.
• It can only be used when you know the value of the function at an even number of equally spaced points.
• Errors in Simpson’s Rule can still occur, particularly when the function is not well-approximated by a quadratic function over the intervals chosen.

Understanding and applying Simpson’s Rule is a key part in approximation of definite integrals, and will be valuable in problems where analytical methods are challenging or impossible. Remember to check the limitations of the rule before applying it as an approximation method.