Question

11. value 0.00 points Identify the estimates of the the given integralusing the two-, three-, and four-point Gauss-Legendre formulas Two-polnt formula-406.295 Two-point formula-5041305 Three-point formula-495.8204 Three-point formula-406.295 Four-point formula-5041305 Four-point formula-495.8204

Answer 1

F (x) = integral_0^3 x e^2x dx gauss 2 - point: - integral_a^b f (x) dx c_1 f (x_1) + c_2 f (x_2) let f (x) = x e^2x, a = 0, b = 3 c_1 = b - a/2 = 3- 0/2 = 3/2 = 1.5 c_2 = b - a/2 = 3 - 0/2 = 3/2 = 1.5 x_1 = b - a/2 (-1/squareroot 3) + b + a/2 = 3 - 0/2 (-1/squareroot 3) + 3 + 0/2 = -0.8660 + 1.5 = 0.634 x_2 = b - a/2 (1/squareroot 3) + b + a/2 = 0.8660 + 1.5 = 2.366 f (x_1) = 0.634 (e^2 (0.634)) = 0.634 (e^1.268) = 0.634 times 3.554 = 2.2532 \r\n f (x_2) = 2.366 (e^2 (2.366)) = 2.366 (e^4.732) 2.366 times 113.522 = 268.593 integral_0^3 x e^2x dx = c_1 F (x) + c_2 f (x_2) = 1.5 (2.2532) + 1.5 (268.593) = 3.3798 + 402.8895