Advanced Calculus fi..

Chapter 4 Integral Calculus of Functions of Several VariablesThe integral (4.24) is called an elliptic integral of first kind. Integrals of thesecond and third kinds are illustrated bywhere 0 < k2 < 1, a # 0, a2 # k2. The integral (4.25) arises in computing thelength of an arc of an ellipse (Problem 4), and this is the basis of the term ellipticintegral.It can be shown that if R(x, y) is a rational function of x and y (Section 2.4)and g(x) is a polynomial in x of degree 3 or 4, then the integralcan be expressed as an elementary function plus elliptic integrals of the first, second,or third kind. Accordingly, numerical evaluation of a few integrals permits preciseevaluation of a broad class of integrals. This puts additional emphasis on the fact thatevery indefinite integral determines a function of x. For each new indefinite integralthat we study and evaluate, a broad class of new functions becomes both numericallyuseful and open to a thorough analysis. Indeed, the trigonometric functions couldhave been dejned in such a manner:defines y = arc sinx and hence x = sin y. This is more awkward than the usualgeometric definition, but all properties can be deduced from such a definition (cf.Problem 3).For further information on elliptic integrals we refer to the books by von Khiinand Biot (Chapter 4) and Whittaker and Watson (Chapters 20-22) listed at the endof the chapter. Some properties are obtained in Problems 5 to 7.PROBLEMS b.1. Evaluate numerically:a) JxdxforO~x< 10,usingx=0,1,2 ,....b) ~e-x2dxfor~_(x 5 1,usingx =0,0.1,0.2 ,....C) Jcosxdx, Osx I 1,usingx =0,0.1,0.2 ,....d) J-&dx, Osx 5 1,usingx=O,O.1,0.2 ,....=;l+xzfe) J m d x , O~x~0.5,usingx=0,0.1,0.2 ,....The answers should be graphed and checked by differentiation.