Mean Value Theorem Of Integrability

Mean Value Theorem Of Integrability :

  First Mean Value Theorem :

         If a function f is continuous on [a,b] then ∃ a number ξ in [a,b] such that 

        b

        ∫ f dx = f(ξ) (b - a)

        a

f is continuous , therefore f ∈ R on [ a,b ] .

Proof :

         Given that function f is continuous on  [ a, b] . Let m , M be the infimum and supremum of f in [ a,b ] . Then clearly       we have

                                     b

                m( b - a ) ≤ ∫ f dx ≤ M( b - a )

                                    a

So , ∃ a number μ ∈ [ m, M ] such that 

         b

         ∫ f dx = μ( b - a ) ........(1)

         a

As f is continuous on [ a,b ] , it attains every value between its bounds m and M . 

So ∃ a number ξ ∈ [ a,b ] such that 

                     f(ξ) = μ .........(2)

   Using equations (2) in (1) , we get 

                       b

                      ∫ f dx = f(ξ) (b - a ) 

                      a

Hence the proof .

The Generalised First Mean Value Theorem :

       If f and g are integrable on [ a,b ] and ∃ a number μ lying between the bounds of f such that 

                          b                 b

                          ∫ fg dx = μ∫ g dx

                          a                 a

Proof : 

           Let us suppose that g be positive over [ a,b ] and m , M be the infimum and supremum of f . Now for all x ∈ [ a,b ]        we have  m ≤ f(x) ≤ M 

   ⇒ m g(x) ≤ f(x) g(x) ≤ M g(x)

          b                  b                            b

So m ∫ g(x) dx ≤ ∫ f(x) g(x) dx ≤ M∫ g(x) dx

          a                 a                             a

                                             when b ≥ a

Let μ be a number lying between m and M , then 

         b                  b

         ∫ fg dx = μ ∫ g dx       (Proved)

         a                  a