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