Fubini's Theorem
Fubini's Theorem :- In the world of mathematics integration plays an important role . So we have discussed the advance form of integration which results the famous theorem Fubini's Theorem. Statement: If a double integral , I = ∫∫ f dx dy R exists over a rectangle R = [a,b;c,d] , and if d ∫ f dy also exists , for each fixed x in [a,b] , c b d then the iterated integral ∫ dx ∫ f dy exists a c and is equal to double integral I. Proof : Let ε be any positive number. Since the upper integral , Iᵘ is the infimum of the upper sums , there exists a partition P of R such that Σ Σ Mᵢⱼ ΔRᵢⱼ < Iᵘ + ε i j or Σ Σ Mᵢⱼ Δxᵢ Δyⱼ < Iᵘ + ε ...........(1)