Definitions Related To Riemann Integral

Definitions Related To Riemann Integral :

Partition Of A Closed Interval :

            By partition of [a,b] , we mean a finite set P of points x₀,x₁,x₂.......,xₙ  where 

 a= x₀≤x₁≤x₂.....≤xₙ₋₁≤xₙ = b . Here [x₀,x₁],[x₁,x₂],......[xᵢ₋₁,xᵢ],....[xₙ₋₁,xₙ] are the sub intervals of [a,b] .

       The ith sub interval is denoted by the symbol Δxᵢ and is given by 

       Δxᵢ = xᵢ - xᵢ₋₁ , where i = 1,2,3.....,n

Here Δxᵢ is also stands for the length of the interval .

Definition Of Upper / Lower Sums :

           Let f is a bounded real function on [a,b]. Then f is bounded on each sub interval corresponding to each partition P . Let Mᵢ and mᵢ be the supremum and infimum of the function f corresponding to P in Δxᵢ .

       Then the upper and lower sums of f corresponding  to P are given by
                n
 U(P,f) = ΣMᵢΔxᵢ =M₁Δx₁ + M₂Δx₂ + .....+MₙΔxₙ
               i=1

and
              n
L(P,f) = Σ mᵢΔxᵢ = m₁Δx₁ + m₂Δx₂ + .....+mₙΔxₙ
              i=1

Where U(P,f) = upper sums
              L(P,f) = lower sums

Definition Of Upper / Lower Integrals :

                The upper and lower integral of the function f over the [a,b] are given by

  - b

  ∫f dx = inf U or inf U(P,f) ∀ partition P

   a

               and

    b

   ∫ f dx = sup L or sup L(P,f) ∀ partition P .

  - a

Definition Of Darboux's Condition Of Integrability :

           The function f is said to be Riemann Integrable (Integrability) over [a,b] if 

       b            b          - b

      ∫ f dx = ∫ f dx = ∫ f dx 

       a         - a            a

Oscillatory Sum :

            The oscillation of the function f in the sub interval Δxᵢ , with respect to the partition P is denoted by W(P,f) and is given by 

             W(P,f) = U(P,f) - L(P,f)

and is non _ negative .

Definition Of Norm :

                  For any partition P , the length of the largest sub interval is called the norm or mesh of the partition and is denoted by μ(P) or μ . So μ(P) = max Δxᵢ  ( 1≤ i ≤ n ) 

Definition Of Refinement : 

         A partition P* is said to be a refinement of P if P ⊂ P* i.e every point of P is a point of P* .

Definition Of Common Refinement :

         If P₁ and P₂ are two partitions , then their common refinement is given by 

           P*  = P₁ ∪ P₂  .

Definition Of Riemann Sums :

           A Riemann sum if f over [ a,b ] relative to a partition P of [ a,b ] is denoted by S(P,f) and is given by 

                       n

         S(P,f) = Σ f(tᵢ) Δxᵢ   where t₁,t₂,...tₙ 

                      i=1

points such that xᵢ₋₁≤ tᵢ ≤xᵢ   (i = 1,2,..n)

Second Definition Of Integrability :

             A function f is said to be integrable on [a,b] if lim ≤ (P,f) exists as μ(P) --->0

and then   

                                             b

                    lim  S(P,f)   = ∫   f  dx 

                μ(P)-->0               a

About Scientist :

           

       Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series.