Page - 133 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

6.1 Basic IdeasofNumerical Integration 133 where h= b−a n . (6.4) That is, we getn sub-intervalsof the same sizeh. Given the integrationpoints, the original integral is re-written as a sum of integrals, each integral being computed over the sub-interval between two consecutive integration points. The integral in (6.2) is thusexpressedas ∫ b a f(x)dx= ∫ x1 x0 f(x)dx+ ∫ x2 x1 f(x)dx+ . . .+ ∫ xn xn−1 f(x)dx. (6.5) Note thatx0 =a andxn=b. Proceeding from (6.5), the different integration methods will differ in the way theyapproximateeach integralon the righthand side. The fundamental idea is that each termisan integralovera small interval [xi,xi+1], andover this small interval, itmakessense toapproximatef byasimpleshape,sayaconstant,astraight line,or aparabola, thatcanbeeasily integrated.Thedetailswillbecomeclear in thecoming examples. Computational Example To understand and compare the numerical integration methods, it isadvantageoustouseaspecificintegralforcomputationsandgraphical illustrations. In particular, we want to use an integral that we can calculate by hand such that the accuracy of the approximation methods can easily be assessed. Ourspecific integral is takenfrombasicphysics.Assumethatyouspeedupyour car from rest, on a straight road, and wonder how far you go in T seconds. The displacement is given by the integral ∫T 0 v(t)dt, where v(t) is the velocity as a functionof time.A rapidly increasingvelocity functionmightbe v(t)=3t2et3 . (6.6) Thedistance traveled in1s is then ∫ 1 0 v(t)dt, (6.7) which is the integralweaim tocomputebynumericalmethods. By hand,weget ∫ 1 0 3t2et 3 dt= [ et 3 ]1 0 ≈1.718, (6.8) which is roundedto 3decimals forconvenience.