数值分析-12-Integration.pdf

12. umerical Integration 12.1 Introduction 12.1.1 Differentiation and Integration Calculus is the mathematics of change. Because engineers must continuously deal with systems and processes that change, calculus is an essential tool of engineering. Standing in the heart of calculus are the mathematical concepts of differentiation and integration. y f (x + x) − f (x ) dy f (x + x) − f (x ) = i i , = lim i i x 0 x x dx x b I = ∑f (x )x = f (x)dx i ∫ x →0 a The functions that we need to differentiate or integrate will typically be in one of the following three forms: • A simple continuous function such as a polynomial, an exponential, or a trigonometric function. we can evaluate the derivative or integral analytically using calculus. • A complicated continuous function that is difficult or impossible to differentiate or integrate directly. need to use numerical methods • A tabulated function where values of x and f(x) are given at a number of discrete points, as is often the case with experimental or field data. need to use numerical methods 12.1.2. Graphical representation of integration: According to the dictionary: the meaning of integration is the total value or summation of f(x)dx over the range x=a to b. The integral corresponds to the area under the curve of f(x) between x=a and b.