Outline
                                   Introduction and Objective
                         Riemann Integrals and Differentiation
                       Integration of Vector Valued Functions
                                                  References
                  Riemann-Stieltjes Integrals - Integration and
                                                 Differentiation
                                                Dr. Aditya Kaushik
                                           Directorate of Distance Education
                           Kurukshetra University, Kurukshetra Haryana 136119 India
   Dr.A.Kaushik: Lecture-4 Real Analysis M.Sc.-I (Mathematics)    Directorate of Distance Education, K.U. Kurukshetra
                                                      Outline
                                   Introduction and Objective
                         Riemann Integrals and Differentiation
                       Integration of Vector Valued Functions
                                                  References
                  Introduction and Objective
                  Riemann Integrals and Differentiation
                  Integration of Vector Valued Functions
                  References
   Dr.A.Kaushik: Lecture-4 Real Analysis M.Sc.-I (Mathematics)    Directorate of Distance Education, K.U. Kurukshetra
                                                      Outline
                                   Introduction and Objective
                         Riemann Integrals and Differentiation
                       Integration of Vector Valued Functions
                                                  References
                  Till now we dealt with some of the properties of the
                  Riemann-Stieltjes integrals.
                  In this lesson, we make an attempt to exploit our knowledge
                  of derivatives to compute the integrals.
                  The results we prove can be seen as a useful tool that
                  provides us some of the nice properties that we seek.
   Dr.A.Kaushik: Lecture-4 Real Analysis M.Sc.-I (Mathematics)    Directorate of Distance Education, K.U. Kurukshetra
                                                      Outline
                                   Introduction and Objective
                         Riemann Integrals and Differentiation
                       Integration of Vector Valued Functions
                                                  References
                  Westart with the Mean-Value theorem for integrals, which
                  reads
          Theorem
          Suppose that f is continuous on I = [a;b], then there exists a
          number s ∈ I such that
                                           Z b f (x)dx = f (s)(b − a):
                                             a
   Dr.A.Kaushik: Lecture-4 Real Analysis M.Sc.-I (Mathematics)    Directorate of Distance Education, K.U. Kurukshetra