Several rules for approximating the value of an indefinite integral were formulated. To recall, we have Riemanns Sums, the Trapezoidal Rule, the fundamental theorem of calculus, the concept of limit, and many others. Now, let us study another approximation of the integral with the use of Simpson’s rule.  Simpson’s rule is founded on a quadrature (quadratic) form of the function to be integrated. This is indeed in contrast to the trapezoidal form under trapezoidal rule. Nevertheless, the underlying principle in this rule is the same as that of the Riemanns sums and the trapezoidal rule. Suppose a continuous function y=f(x) is defined over the interval [a,b].  Once more, we introduce by partitioning [a, b] into sub-intervals all of the same distance, but at this point we should apply an even number of intervals, so n will be even.

The same with the trapezoidal rule, the main objective of the Simpson’s rule is to approximate the area bounded by the curve over an interval [a,b] and the x–axis by partitioning the region into strips by somewhat more complex figure than a rectangle. At this point, we will form a parabolic figure. We consider the strips in pairs, (so n is even) and sketch a parabola through the three points of divisions. The three points must have the abscissas x = a, x = ½ (a + b) and x = b while the corresponding ordinates must result from the evaluation of f(x) at these abscissas, so we have f(a), f((a+b)/2) and f(b), respectively. Precisely, the area of the region underneath this parabolic curve based on the prismoidal formula is equal to one–third of delta x (the change in x-values from one point of division to the succeeding point) times the summation of the corresponding ordinates (value of the function) at x=a, x=b and four times the ordinate of x=½(a+b).  In this case delta x is obtained by getting half of the difference of the limits, a and b. This formula is then the basis for Simpson’s rule.

Simpson’s Rule is making use of the equation of the parabolic figure formed in finding the area of the region. Generally, the Simpson’s Rule states that if the number of strips, n is even, and, as before, xk = a + kx = a + k (b-a)/n, then the indefinite integral of the function, y = f(x) over the interval [a,b] is approximately equal to the product of the difference of the limits, b and a divided by 3n and the summation of [f(x0) + 4 f(x1) + 2 f(x2) + 4 f(x3) + ... + 2 f(xn – 2) + 4 f(x n – 1 ) + f(xn)]. The divisor 3 indicates the minimum requirement of equal – length fragments.

Again, we can lengthen this approximation to higher precision by partitioning our target interval domain into as many n equal-length fragments. The area of the region underneath the parabola and the lines at x=a and x=b, is then equal to the weighted sum of the above formula for each match up of adjacent regions, which best works out for even n.  

Simpson's Approximation Example Problem