Newton_Cotes Quadrature Formula
Newton_Cotes Quadrature Formula: b Let I = ∫ f(x) dx ............(1) a Where f(x) takes the values y₀,y₁,y₂.....yₙ for x= x₀,x₁,x₂.........xₙ Let us divide the interval (a,b) into n sub_intervals of width h so that x₀=a , x₁= x₀+h , x₂=x₀+2h ......xₙ=x₀+nh=b x₀+nh Then I= ∫ f(x) dx Put x=x₀+rh ⇒dx =hdr x₀ n = h ∫ f(x₀+rh) dr 0 n = h ∫ [y₀+rΔy₀+r(r-1)/2! Δ²y₀ 0 + r(r-1)(r-2)/3! Δ³y₀ + r(r-1)(r-2)(r-3)/4! Δ⁴y₀ + r(r-1)(r-2)(r-3)(r-4)/5! Δ⁵y₀ + r(r-1)(r-2)(r-3)(r-4)(r-5)/6! Δ⁶y₀ +.....]dr [Using Newton's Forward Interpolation Formula] Now integrating term by term we get x₀+nh ∫ f(x) dx = nh[y₀+n/2 Δy₀ +n(2n-3)/12 Δ²y₀ x₀ + n(n-2)²/24 Δ³y₀