Four beautiful quadrature rules
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A framework is presented to compute approximations of an integral I(f)=∫_a^b 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=∫_0^π 2 sin^2(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.
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