Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY CALCULUSf(x)ax 1 x 2 x 3 x 4 x 5ξ 1 ξ 2 ξ 3 ξ 4bxFigure 2.8 The evaluation of a definite integral by subdividing the intervala ≤ x ≤ b into subintervals.◮Evaluate from first principles the integral I = ∫ b0 x2 dx.We first approximate the area under the curve y = x 2 between 0 and b by n rectangles ofequal width h. If we take the value at the lower end of each subinterval (in the limit of aninfinite number of subintervals we could equally well have chosen the value at the upperend) to give the height of the corresponding rectangle, then the area of the kth rectanglewill be (kh) 2 h = k 2 h 3 . The total area is thus∑n−1A = k 2 h 3 =(h 3 ) 1 n(n − 1)(2n − 1),6k=0where we have used the expression for the sum of the squares of the natural numbersderived in subsection 1.7.1. Now h = b/n and so( ) (b3 n b3A = (n − 1)(2n − 1) = 1 − 1 )(2 − 1 ).n 3 6 6 n nAs n →∞, A → b 3 /3, which is thus the value I of the integral. ◭Some straightforward properties of definite integrals that are almost self-evidentare as follows:∫ ba∫ ca∫ ba0 dx =0,f(x) dx =∫ b[f(x)+g(x)] dx =a∫ aaf(x) dx +∫ ba60f(x) dx =0, (2.23)∫ cbf(x) dx +f(x) dx, (2.24)∫ bag(x) dx. (2.25)