Computing Integrals

Abstract

We now turn our attention to solving mathematical problems through computer programming. There are many reasons to choose integration as our first application. Integration is well known already from high school mathematics. Most integrals are not tractable by pen and paper, and a computerized solution approach is both very much simpler and much more powerful – you can essentially treat all integrals R b a f .x/dx in 10 lines of computer code (!). Integration also demonstrates the difference between exact mathematics by pen and paper and numerical mathematics on a computer. The latter approaches the result of the former without any worries about rounding errors due to finite precision arithmetics in computers (in contrast to differentiation, where such errors prevent us from getting a result as accurate as we desire on the computer). Finally, integration is thought of as a somewhat difficult mathematical concept to grasp, and programming integration should greatly help with the understanding of what integration is and how it works. Not only shall we 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 to write Python code that is free of programming mistakes. Calculating an integral is traditionally done by b