2019/06/14 by Graça, Mário M.
#65-05 #65D30 #65D32 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1906.06124
A framework is presented to compute approximations of an integral I(f)= ∫ab f(x) dx from a pair of companion rules and its associate rule. We show that an associate rule is a weighted mean of two companion rules. In particular, the trapezoidal (T) and Simpson (S) rules are weighted means of the companion pairs (L,R) and (T,M) respectively, with L the left rectangle, R the right rectangle and M the midpoint rules. As L,R,T and M reproduce exactly the number π= ∫0π2 sin2(x) dx, we named them the four "beautiful" rules. For this example the geometrical interpretation of the rules suggest possible applications of the transcendental number π in architectural design, justifying the attribute beautiful given to the mentioned rules. As a complement we consider other appropriate integrand functions f, applying composite rules in order to obtain good approximations of π, as shown in the worked numerical examples.