Trapezoidal Rule
Trapezoidal Rule : Let us suppose that the interval (a,b) be divided into n equal subintervals 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