Question

2. Use the Trapezoidal Rule to numerically integrate the following polynomial from a tob 1.5 f(x) = 0.2 + 25x-200x' + 675xs_ 900x. + 400x5 Use three different numbers of segments (n- 2, n 3, and n 4) and show the effect of the number of segments on relative error considering the exact value of the integral which is 48.7719. (Note: Show the steps of calculation).

Answer 1

Given, f (x) = 0.2 + 25x - 200x^2 + 675 x^3 - 900 x^4 + 400 x^5 where a = 1 b = 1.5 & integral^b_a f (x) dx = 48.77 w eactvolw For n = 2 By trapezoidal rule. because integral^b_a f (x) dx = h/2 [f (0) + 2f (a +h) + 2f (a + 2h) + ... + f (b)} where h = b - a/n So, here, h = b - a.n = 1.5 - 1/2 = 0.5/2 = 0.25 So, f (a) = f (1) = 0.2 + 25 (1) - 200 (1)^2 + 67.5 (1)^3 - 900 (1)^4 + 400 (1)^5 f (1) = 0.2 f (a + h) = f (1 + 0.25) = f (1.25) = 0.2 + 25 (1.25) + 200 (1.25)^2 + 675 (1.25)^3 - 900 (1.25)^4 + 400 (1.25)^5 f (1.25) = 60.746875 f (b) = f (1.5) = 0.2 + 25 (1.5) - 200 (1.5)^2 +