Page - 136 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

136 6 ComputingIntegralsandTestingCode on the righthandside witha single trapezoid. Indetail, ∫ b a f(x)dx= ∫ x1 x0 f(x)dx+ ∫ x2 x1 f(x)dx+ . . .+ ∫ xn xn−1 f(x)dx, ≈hf(x0)+f(x1) 2 +hf(x1)+f(x2) 2 + . . .+ h f(xn−1)+f(xn) 2 (6.15) Bysimplifying the righthandsideof (6.15)weget ∫ b a f(x)dx≈ h 2 [f(x0)+2f(x1)+2f(x2)+ . . .+2f(xn−1)+f(xn)] (6.16) which ismorecompactlywrittenas ∫ b a f(x)dx≈h [ 1 2 f(x0)+ n−1∑ i=1 f(xi)+ 1 2 f(xn) ] . (6.17) CompositeIntegrationRules The word composite is often used when a numerical integration method is appliedwith more than one sub-interval.Strictly speaking then, writing, e.g., “the trapezoidal method”, should imply the use of only a single trapezoid, while “the composite trapezoidal method” is the most correct name when several trapezoids are used. However, this naming convention is not always followed, so saying just “the trapezoidal method” may point to a single trapezoidaswell as thecomposite rulewithmanytrapezoids. 6.2.2 AGeneral Implementation Specific or General Implementation? Suppose we want to compute the specific integral ∫1 0 v(t)dt, wherev(t)= 3t2et 3 , using the (composite) trapezoidal method in (6.17). Although simple in principle, the practical steps are often confusing to many, because the notation in the abstract formulation in (6.17) differs from the notation in our special problem. Clearly, thef ,x, andh in (6.17) correspond tov, t, andperhapsΔt for the trapezoidwidth inourspecial problem.