Seite - 135 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Band Second Edition

6.2 TheCompositeTrapezoidalRule 135 Fig. 6.2 Computing approximately the integral of a function as the sum of the areas of the trapezoids h3 = (0.8−0.6), (6.12) h4 = (1.0−0.8) (6.13) With v(t) = 3t2et3, each term in (6.9) is readily computed and our approximate computationgives ∫ 1 0 v(t)dt≈1.895 . (6.14) Compared to the true answer of 1.718, this is off by about 10%. However, note that we used just four trapezoids to approximate the area. With more trapezoids, the approximation would have become better, since the straight line segments at the upper trapezoid side then would follow the graph more closely. Doing another hand calculation with more trapezoids is not too tempting for a lazy human, though, but it is a perfect job for a computer! Let us therefore derive the expressions for approximating the integral by an arbitrary number of trape- zoids. 6.2.1 TheGeneralFormula For a given function f(x), we want to approximate the integral ∫b a f(x)dx by n trapezoids (of equal width). We start out with (6.5) and approximate each integral