Numerical Integration - Simpson's Rule
Numerical integration is an operation that many older machines are not able to perform without add-on libraries (e.g. Math Pac for HP-41C) or without writing a program from scratch to do it. There are, of course, some exceptions to this; Hewlett Packard's HP-15C and HP-42S come straight to mind, as does the Casio fx-180P, for example, but these machines are actually the exception, not the rule. There are many methods used to calculate definite integrals, that is integrals between known boundaries yielding a numerical result. One method that is a good trade-off between precision and complexity of application is Simpson's Rule , which attempts to find a quadratic of the form $a\cdot x^2+b\cdot x+c$ that fits or is close to our function. Finding the integral of this polynomial is very easy: \[ \int_{x_1}^{x_2} \! (a\cdot x^2 + b\cdot x + c) \, \partial x = \frac a 3 (x_2^3-x_1^3) + \frac b 2 (x_2^2-x_1^2) + c\cdot (x_2 - x_1) \] Basically, Simpson's Rule states: \[ ...