Question

Alpha=9 beta=3 yazarsin

3. ( 15p.) Approximate the following integral using the two-point Gaussian quadrature rule À (x + a)?e(x-1)2-3 da 4. Consider the following system.

Answer 1

ANSWER

0 solution: We have to approximate the following integral using the two- point Gaussian quadrature role The integral is: ficutd]e C4-1) - Pax Since <= 98 B = 3 Integral becomes. C++89)%2C4-12-20 let IPJ (+932€ C2-1123d I let f(x) = (x +97² CK-1) ²3 Now, The two-point Gaussian quadratia 2 de 0 2 de. o b rule for Secuide is b s f(n) dhe 0 ºC)+ Caft) a where c, = 6-9 & Gz = b-a 2 x = b-a = 6-a ( 7 ) + b ta Az x 2 = 6-0 (i) Now for our case, we have - $() = (x +9)&cx-12-3 - 6-0 (1) + bta

and a=o, b=2. c=2-0 1 C-1 :|C2= 1 15" less)+(2+) -It V3 = 1- V3 : 3 - 1 0.73205080756 = 0.49265 13 and "a = 11 1+1 -1 +13 3 2.73205080757 = 1.57735 13 50 f(0) = f (0.49265) - (0.42265 +9) e (0.42265-12-3

3 =6.42265) € (-0.59735)23 = 88.78 63330 225 0.3333330225-3 = 88.7863330225e- -2.6666669775 = 88.7863330225x0.069 48342462. = 6.16917892223 and f (222) f (20.) = 6:16917 f(1.57735) - (1.57735+9)< (1.57735-1)23. -(10.577359260.5435)23 , = 111.880333022 x <0.3333330225) -3 -2,666 666.9775 111.88 0.33 024x e 111.880333022 x0.06948 344 2962 - = 7.77382924596. f (42) = 7.77382

Hence from two - point Gauss quadratuee rules 2 (4+972 e (K-1) ²3 da ~ Gif (",) + Caf (362) ñ 116.16917)+1(9.7.77382) ñ (6.16917+7.7738) 13.9 4297 Hence the approximate value of the given integral is jot s x +9) ² (M-1)=3 dx ~ 13.94297