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 )