Simpson's One _ Third Rule
Simpson's One_Third Rule : Let us suppose that interval (a,b) be divided into n equal sub intervals such that a = x₀,x₁,.........,xₙ₋₁,xₙ = b We know from Newton's Forward Difference Interpolation formula that y = y₀+ sΔy₀+s(s-1)/2! Δ²y₀+s(s-1)(s-2)/3! Δ³y₀ + ................ .........(1) Where xₙ = x₀+nh and x= x₀+sh .......(2) Integrating both sides of equation (1) , between x₀ and xₙ we get xₙ xₙ ∫ y dx = ∫ (y₀+sΔy₀+s(s-1)/2! Δ²y₀ x₀ x₀ + s(s-1)(s-2)/3! Δ³y₀+.......)dx Using equation (2) in the above expression , we get xₙ n ∫ y dx = h∫ (y₀+sΔy₀+s(s-1)/2! Δ²y₀ x₀ 0 +s(s-1)(s-2)/3! Δ³y₀+.....)ds xₙ or, ∫ y dx = nh[y₀+n/2 Δy₀+n(2n-3)/12 Δ²y₀ x₀ + n(n-2)²/24 Δ³y₀+........