Countable and uncountable sets. Riemann integral. Integrability of continuous and monotonic functions. The fundamental theorem of integral calculus. Mean value theorems of integral calculus. Improper integrals and their convergence, Comparison tests, Abel and Dirichlet tests. Beta and Gamma functions. Frullani integral. Integral as a function of a parameter. Continuity, derivability and integrability of an integral of a function of a parameter.
Double and triple integrals. Fubini Theorem without proof, Change of order of integration in double integrals, volume of a region in space, Triple integrals in spherical and cylindrical coordinates, substitution in multiple integrals.
Vector Integration, Gauss, Divergence and Green Theorems.
Sequences and series of functions, pointwise and uniform convergence, Cauchy criterion for uniform convergence, Weierstrass M-test, Abel and Dirichlet tests for uniform convergence, uniform convergence and continuity, uniform convergence and Riemann integration, uniform convergence and differentiation, Weierstrass approximation theorem, Abel and Taylor theorems for power series.Fourier Series. Fourier expansion of piecewise monotonic functions.