Question

3. Use the Composite Trapezoidal rule to estimate the integral 67e**dx by taking the 11 nodes. 4. Evaluate the integral foodx by using composite Simpson's įrd rule with spacing h = 0.25. Compare the error with the exact value. 5. Compute the value of 1.2 (sin r – log I +e")dx by using the composite Simp- son's th rule with spacing h = 0.2.

Please attempt only if you know the solution.
I have received wrong solutions many times.

Answer 1

Roll I= %=0 xn=2 We need n=10 given integral into lo interval, then (3) Jexe dx to take 11 nodes, ie 10 intervals, And if we opply trapezoidal on each interval, we call it composite trapezoidal rule. no. of intervals for an integral J foxida, trapezoidal formula is given by [ f(b)+fca] h [ f(b) + f(a) apply trapezoidal rule on each interval. h = xn-% 2-0 lo = 0.2 η h=0.2 b b b-a J f (x) 2 a we divide

+2.ellje so I- j ex'de 0.4 06 2.0 و زه dx + J et de + J er det + Sexdx 0.2 0.4 1.8 on lehet eloase) - on leven lang of the ran acony = 0.1 [ eo +20 (02) (0.42 42. 0(0.6)? (0.8)E + q.e + 9 e +2.0 (1.23² .(1.4)? +2.e +2.e (1.6)2 (1.8) 2 (2) 2 +2.e ta.e 1.e ce? 0.9 [1 + 2 (1.04 +1.174 + 1.433+1.896+ 2.718 +4.22 +7.099+ 19.935 + 95.533)+54 598 0.1 [1+ 2 (58.048) +54-598) 0.1[171.694] 17.17 (abbrox) 11 - I = 17.17 Any

(4) I= 2 2. da 1 +23 h=0.25 11 b-a = 0.25 - n=1 - 4 =) (n=4 0.25 no, of intervals Composite simbson 1/3 formule [ f(46) + 446628,7 ++(43) +--- FC Am-,) 246(22) + f(x2)+ .+ f (Xn-2) ? + teen] + Iz o - dx o lt23 = h [ f(26) + 4 ( f(2,) + f(a)) + 2 (f (22)) + f(24)] = 0.25 OT (460) + u (f(0.25)+ f(0,75))+ 9.4(0.5)+fo] = 0.25 [ 0.0625 +0.015625 + 0.5625 1+0.421875 ) + ?.0.15 1+0.125 it 0 95 [4(0.0615 +0.3956) + 0.444 +05] 1+)

- 0.25 [ 1.8284 + o.guu 3 0.25 -X 2.772 3 = 0.231 approx so I= 0.231 approin Now we find exact value of I I = da 382 .da 1+23 3 1+ 3 į (inctx"}] 1/3 [and-lni - I exact ena 3 0.23105 error Texact - I approx =0.23105 -0.231 0.00005 error is negligible by Iz simpson method

sol (5) I= J isina- logx tex) da 1.4 h=0.2 1.4-0.2 0.2=1.2 n=6 composite simpson formula n-2 ni n-3 I= 34 [ +0%) + 3 [ +(2,1 +3E(:) + 25 +(7:) + f(7n f(n)] [=1,4,70 (=2,5,8.. i= 3,6,9 .. so here 9.4 I= J (sina-logy te alda 0.2 3h [ +(0.2) + 3(+(4,1)+(94))+3 (f(x) + +(2,5)) + f(x)] + 2 f(%) 31 [ 4102)+3 (f(0.4)++(1)) +3 (+(06) + f(1 2)) +2 +603) + f(1.4) = 3h 8 [ 3.03 + 3 (2.76+3.56) + 3 (2.90+4.07)+2x3.17 +4.70

0.283 8 (3.03 + 19.02 + 20.91 +6.34 +4.7] 0.6 Xsu =a. 4.05 8 =) I=4.05 by composite simpson / rule
If you find any difficulty in understanding or maybe if you find any calculation mistake, please let me know in the comments section.Thanks!