Mathematical Methods for Physics and Engineering - Matematica.NET

2.2 INTEGRATIONf(x)abxFigure 2.7An integral as the area under a curve.2.2 IntegrationThe notion of an integral as the area under a curve will be familiar to the reader.In figure 2.7, in which the solid line is a plot of a function f(x), the shaded arearepresents the quantity denoted byI =∫ baf(x) dx. (2.21)This expression is known as the definite integral of f(x) betweenthelower limitx = a and the upper limit x = b, andf(x) is called the integrand.2.2.1 Integration from first principlesThe definition of an integral as the area under a curve is not a formal definition,but one that can be readily visualised. The formal definition of I involvessubdividing the finite interval a ≤ x ≤ b into a large number of subintervals, bydefining intermediate points ξ i such that a = ξ 0