Question

Question 1 Consider the following integral: (-0.055x* +0.86x? - 4.2x2 + 6.3x + 2)dx a) Evaluate the integral analytically. (Round the final answer to four decimal places.) b) Use Romberg integration to evaluate the integral to an accuracy of Es=0.5%. Calculate bother and Eg for each level (i.e. for each value, 176, as shown in the class notes) and stop calculating integral estimates once Eg SE. (Round the final answer to three decimal places.) c) Evaluate the integral using the three-point Gauss quadrature formula. (Round the final answer to three decimal places.) d) Evaluate the integral in MATLAB using the integral function. (Round the final answer to three decimal places.)

just c please

please I only need the solution for c

Answer 1

Solution 1.. L e). The giver Integral is LE(-0.055** + 0.86 x°- 4.2.x*+ 6.32 + 3) d2 This problem is of * Gaussian. Three integral into by. Numerical methods. point formula converts its linear Transform an given formula. The Above is the quadrature Here, 12 ¢ (baajt + Chitaj where, a=0.2 b = 8. | 2 = (8-0) + + +68+) x = 4x + 4

Now, using the limits, At X=0 L _0= 4404 x = 8 8 = 4++4 4+ = 4 t = 1 1 - -.--.-.- 44x) = f(t), dx=dt. we have, x=44+4 .: Here, f(x) = (-0.055 x 7.0.86x=4.2 x++. *2)=f(t) = 44144 is f(t) = (-0,055( 4t+43°+ 0,86(44+9) - 4.2(44+49% $13(40+42+2) 1- MIMARRRAAMATULTITILS 1- 1- - - .055 x' + 0.86** - 4.2x?+ 6.32 +2) dx = -0.05s (44+4)* +0.86 (4x +4)*- 4.2 (44 +9327 6.3 (44+4) + 2) at :'

Now, Using Quadrature the Gauss formula. Ihree point 366) = 100 * OF Equation ? f(0) = -0.05564807439 + 0.868480+493-4.2(4X0+4)* -' + 6.364x0+43 + 2 Ife - 0.964 Now, s 3.0. 055 (4x = F +4)* + 0.88(4X=f+) -4.2(4x+4734 6:31435* + 4) + 2 a Hina) = 4.85992 ff ) = -0.08s (4x124) + 0.88(415+4) |--4.2( 049.63(4*+9) + 2. F E) = 4.85717 = 0.055

Substituting the and Fe values of f(o), f(-) in eq-0. + 5 14.8599 +4.8599 olo 99) + $ x 9.7198 0,8533 + 5.3998 - 6,25313 Quadrature Hence, By using the 'Three-point formula , we get . . RAAANAAMAAN I = 6.25 313 og 6.253 Hence, This is the Solution required