Basic Ideas and Definitions Definite and Indefinite Integrals  TheFundamentalTheoremofCalculus BasicTechniquesofIntegration IntegralsInvolvingPartial Fractions Integration by Parts Integration by Substitution Integration of More Complicated Trigonometric Functions
                                       Engineering Analysis 1 : Integration
                                     Dr. Paul D. Ledger, Dr Rob Daniels, Dr Igor Sazonov
                                                             engmaths@swansea.ac.uk
                                                     College of Engineering, Swansea University, UK
                                               PDL,RD,IS (CoE)                                                           WS2016      1/ 39
    Basic Ideas and Definitions Definite and Indefinite Integrals  TheFundamentalTheoremofCalculus BasicTechniquesofIntegration IntegralsInvolvingPartial Fractions Integration by Parts Integration by Substitution Integration of More Complicated Trigonometric Functions
    Outline
            1    Basic Ideas and Definitions
            2    Definite and Indefinite Integrals
            3    TheFundamentalTheoremofCalculus
            4    Basic Techniques of Integration
            5    Integrals Involving Partial Fractions
            6    Integration by Parts
            7    Integration by Substitution
            8    Integration of More Complicated Trigonometric Functions
                                               PDL,RD,IS (CoE)                                                           WS2016      2/ 39
    Basic Ideas and Definitions Definite and Indefinite Integrals  TheFundamentalTheoremofCalculus BasicTechniquesofIntegration IntegralsInvolvingPartial Fractions Integration by Parts Integration by Substitution Integration of More Complicated Trigonometric Functions
    Basic Ideas and Definitions
            Asyouprobably know, the process of finding areas under the graph of a function is
            called integration.
            Theareaunderthegraphofafunction f(x) is called its integral.
            For simple cases we can work this out from geometry:
                                     f(x)                                   f(x)
                                                                                          f(x)=x
                                                                f(x)=1
                                       1
                                               a       b              x               a       b              x
                                       Area under graph of f(x) = 1 is 1.(b − a) = b − a
                                                                                       1            2     1    2      2
                          Area under graph of f(x) = x is a(b − a) + 2(b − a) = 2(b −a ),
                                               PDL,RD,IS (CoE)                                                           WS2016      3/ 39
    Basic Ideas and Definitions Definite and Indefinite Integrals  TheFundamentalTheoremofCalculus BasicTechniquesofIntegration IntegralsInvolvingPartial Fractions Integration by Parts Integration by Substitution Integration of More Complicated Trigonometric Functions
    Whydoweneedintegrationasengineers?
            Integration has many important applications in engineering, here are just a few:
                   Calculating the centroid of area;
                   Calculating moments of inertia;
                   Calculating the work by a variable force;
                   Theforces due to presence of electrical charges;
                   Force exerted by liquid pressure.
                   ...
            All these applications use the basic techniques we will learn in the coming lectures.
                                               PDL,RD,IS (CoE)                                                           WS2016      4/ 39