Page - 134 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

134 6 ComputingIntegralsandTestingCode 6.2 TheCompositeTrapezoidalRule The integral ∫b a f(x)dx may be interpreted as the area between thex axis and the graphy=f(x)of the integrand.Figure6.1illustrates thisareafor thecase in(6.7). Computing the integral ∫1 0 v(t)dt amounts to computing the area of the hatched region. If we replace the true graph in Fig.6.1 by a set of straight line segments, we may view the area rather as composed of trapezoids, the areas of which are easy to compute. This is illustrated in Fig.6.2, where four straight line segments give rise to four trapezoids, covering the time intervals [0,0.2), [0.2,0.6), [0.6,0.8) and [0.8,1.0]. Note that we have taken the opportunity here to demonstrate the computationswith time intervals thatdiffer in size. Theareasof thefour trapezoidsshowninFig.6.2nowconstituteourapproxima- tion to the integral (6.7): ∫ 1 0 v(t)dt ≈h1(v(0)+v(0.2) 2 )+h2(v(0.2)+v(0.6) 2 ) +h3(v(0.6)+v(0.8) 2 )+h4(v(0.8)+v(1.0) 2 ), (6.9) where h1 = (0.2−0.0), (6.10) h2 = (0.6−0.2), (6.11) Fig. 6.1 The integral ofv(t) interpreted as the area under the graph ofv