Advanced Integral Calculus: Convergence of Integrals, Eulerian Integrals, Multiple Integrals, Volume and Areas of Surfaces

The book offers a second course on Integral Calculus (of functions of real variables) for Graduate and Engineering students. Convergence (including uniform convergence) of improper integrals and various tests of Abel, Dirichlet and Weierstrass for the same are discussed. Improper integrals of quotient functions of various forms are evaluated. Integrations of continuous functions of 2 variables are discussed in relation with differential and integral properties of parameters in the functions. Eulerian integrals: Beta and Gamma functions, their transformation properties, relations connecting them, reflection and duplication formulae for the Gamma function and Frullani's integral are given. Double and triple integrals giving volume and areas of surfaces are discussed in the last 3 chapters. Numerous examples are solved illustrating the methods of change of order of integration. Dirichlet's integrals of 2nd, 3rd and pth orders are evaluated. Transfor- mations of integrals into 2 and 3-dimensional polar coordinates including Dirichlet's and Liouville's integrals are given. A short bibliography of the subject and an alphabetical index are added at the end.