Page - 143 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

6.3 TheCompositeMidpointMethod 143 Fig. 6.3 Computing approximately the integral of a function as the sum of the areas of the rectangles the trapezoidal method (10%) with the same sub-intervals. More rectangles give a betterapproximation. 6.3.1 TheGeneralFormula Let us derive a formula for the midpoint method based on n rectangles of equal width: ∫ b a f(x)dx= ∫ x1 x0 f(x)dx+ ∫ x2 x1 f(x)dx+ . . .+ ∫ xn xn−1 f(x)dx, ≈hf ( x0+x1 2 ) +hf ( x1+x2 2 ) + . . .+hf ( xn−1 +xn 2 ) , ≈h ( f ( x0+x1 2 ) +f ( x1+x2 2 ) + . . .+f ( xn−1+xn 2 )) . (6.19) This sum maybewrittenmorecompactlyas ∫ b a f(x)dx≈h n−1∑ i=0 f(xi), (6.20) wherexi = ( a+ h2 )+ ih.