Question

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You can use the formula for Simpson's Rule given above; but here is a better way. If you already have the Trapezoid Rule approximations T(2n) and T(n), the next Simpson's Rule approximation follows immediately with a simple calculation: 1. S(2n) = 47(2n)-T(n) Verify that for n 8, the two forms of Simpson's Rule are the same. 2. Consider the integral .xx+ 1 dr = 1- 0.2 14601 8366 Compute the Trapezoid Rule and Simpson's Rule approximations to the integral for 64, and fill in the columns labeled T(n) and S(n) in the table below. 4, 8, 16, 32, and Table 1 Tn) S(n) Error T(n) Error S(n) 4 16 32 64

Answer 1

Formula for Sampson's Rule S(2 n) = 4 T (2n) - T(n)/3 ---(1) Given integral integral_0^1 x^2/x^2 + 1 dx = 1 - pi/4 TildeEqual 0.2146018366 where a = 0, b = 1, f(x) = x^2/x^2 + 1, n = 2 h = b - a/n rightarrow 1 - 0/2 = 0.5 x 0 0.5 1 y = x^2/x^2 + 1 0 0.2 0.5 y_0 y_1 y_2 By using Trapezoidal T(2) = h/2[(y_0 + y_1) + 2 y_1] = 0.5/2 [(0 + 0.5) + 2 (0.2)] t(2) = 0.225 At n = 4 then h = 1 - 0/4 rightarrow 1/4 rightarrow 0.25 x 0 0.25 0.5 0.75 y 0 0.058 0.2 0.36 (y_0) (y_1) (y_2) (y_3) To calculate T(4) = h/2 [(y_0 + y_3) + 2 (y_1 + y_2)] = 0.25/2 [(0 + 0.36) + 2 (0.058 + 0.2)] T(4) = 0.1095 At n = 8 then h = 1 - 0/8 = 1/8 rightarrow 0.125