Page - 55 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

3ComputingIntegrals We now turn our attention to solving mathematical problems through computer programming.Therearemanyreasonstochooseintegrationasourfirstapplication. Integration is well known already from high schoolmathematics. Most integrals are not tractable by pen and paper, and a computerized solution approach is both verymuchsimplerandmuchmorepowerful–youcanessentially treatall integralsRb a f.x/dx in 10 lines of computer code (!). Integration also demonstrates the differencebetweenexactmathematicsbypenandpaperandnumericalmathematics on a computer. The latter approaches the result of the formerwithout anyworries aboutroundingerrorsduetofiniteprecisionarithmetics incomputers(incontrast to differentiation,where sucherrorspreventus fromgettinga result asaccurateaswe desire on the computer). Finally, integration is thought of as a somewhat difficult mathematical concept to grasp, and programming integration should greatly help with theunderstandingofwhat integration is andhowitworks. Not only shallwe understand how to use the computer to integrate, but we shall also learn a series of good habits to ensure your computer work is of the highest scientific quality. In particular, we have a strong focus on how towrite Python code that is free of programmingmistakes. Calculatingan integral is traditionallydoneby bZ a f.x/dxDF.b/ F.a/; (3.1) where f.x/D dF dx : 55©TheAuthor(s)2016 S.Linge,H.P.Langtangen,Programming for Computations –Python, Texts inComputational Science andEngineering15,DOI10.1007/978-3-319-32428-9_3