It is a set of methods used to ascertain the numerical value of a definite integral. In other words, it can also be said that it is a method to find out the area inscribed within a given curve with definite end points. [1] There are many methods to determine the area within a continuous function with definite end points. The simplest example is the division of the area into small rectangles, determining the area of rectangles and then integrating all the areas together to determine a resultant area. But the margin of error is huge in this method. This can be minimized by the ‘trapezoidal rule’. [2] The curve is divided into regions of trapezoids and the area is integrated to find out the total area of the curve. The resultant area found has a very less margin of error.
The area within the curve under trapezoidal rule is given by:
Another way of determining the area is the ‘Simpson’s rule’. In this method, we use parabola to determine the area of the curve. It is more efficient than the trapezoidal rule. The given area is divided into n regions of equal area Δx where n is even. [3] Using the value of Δx, different values of y, points on the given curve, are determined. Using these points, the area under the curve is determined.
References
Davis, Philip J. and Rabinowitz, Philip Methods of Numerical Integration, Second Edition Copyright 1984 by Academic Press Inc. Orlando Fld.
Burden, Richard L. and Faires, J. Douglas Numerical Analysis, 8th Edition Copyright 2005 by Thomson Brooks/Cole
Hazewinkel, Michiel, ed. (2001), "Simpson formula", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
