Question

1+4" QUESTION 5 The integral I= is to be approximated numerically. (a) Find the least integer M and the appropriate step size h so that the global error for the composite (10) Trapezoidal rule, given by b-42f"(E), a<<<b, 12 is less than 10-5 for the approximation of I. (b) Use the two-term Gaussian quadrature formula and 6 decimal place arithmetic to approximate I. (Hint: Parameters are g =1, i = 1, 2;rı = 0.5773502692, r2 = -0.5773502692) (9) (c) Evaluate the integral analytically and find the error in the approximation in (a) above. (6) [25]

Answer 1

I= 1 x+4 ennors in componite trapezoidal buke I is given by -(6-2). n {"(t) ALELb 12 Where = shine 06-03 12 n PL obsenve That whene 1515 ke 12 n 16) LEK YXE (6,6) Mon here f(x) = x+4 (S(+4) 2 + x) = (x+4)3 xt(0, 2) Then 2 32 1 f (x) 1 - 32 43 the choose all such that the error is less than ie -5 10 Elk10 thing Hill 10 k (b-a) 2 satisfy it 12 nr 3 2 → žz 32 12.12 .x10 ERR:33 n 7 12 4 45.6435 n 7 48 We choose M = 46.

Then slep size ais choosen he 206357 h 4 2-0 = 2 4 6 2 46 46 to we simplete the can choose A proces of calculation 50 0.04 (b) use the tranformation 01A x + 4 D x=1 tt dx=dt t+5 Then Ano point garren madratune formula, 540) x-f67)+1(5) 고 at 1 -) + +5 +5 W+5 = 0.226 109 +0,179297 = 0.405406 6) 2 x+4 0 2 [borg (* ++) tai lo In 1) - 12 (4) = 0,405465. sedyrenga

for = 46 the erspor lejaL 2 12. (46) 4x/2x (46) AMPAK (103 4 -7 = 9.8 46 x 10 0.0000009846

I= 1 x+4 ennors in componite trapezoidal buke I is given by -(6-2). n {"(t) ALELb 12 Where = shine 06-03 12 n PL obsenve That whene 1515 ke 12 n 16) LEK YXE (6,6) Mon here f(x) = x+4 (S(+4) 2 + x) = (x+4)3 xt(0, 2) Then 2 32 1 f (x) 1 - 32 43 the choose all such that the error is less than ie -5 10 Elk10 thing Hill 10 k (b-a) 2 satisfy it 12 nr 3 2 → žz 32 12.12 .x10 ERR:33 n 7 12 4 45.6435 n 7 48 We choose M = 46.