Question

3) Evaluate the integral ſx cos xdx using the a) Trapezoidal rule and b) Simpson's rule. For each of the numerical estimates, determine the percent relative true errors.

Answer 1

The solution is given below

5 Jacosx Using Trapezoidal Rule, The trapezoidal Rule states that b f(x) dx = su (fo) + 2 f(x) + 2 f(x2) + flan) whol AX - =b-a 1 a-l b=5 and let n=5 7 An= 5-1 24 5 Now, divide the interval [15] into n=5 subinterval of length Dary aal, 17 21 13 5 5-6 5 Now, flxo) = f(a)=41) = cos(1) - 0.540302 2 flos) = 24 2 f19 = 18 cosla cosa سی - 0.817 927

240)= 24(15) = 24 cos (1) = -4.455821 2 H23)> 27 + ( 12 ) = 34 w(17) --6.574 227 241 44) = 24/11 ) = 42 cos (21) --4-118 1908 a flas) = f(b)=45) -swss = 1.4183109 Now, adding up the above values and fulting in Oo, we get 0.540302 + (-0.817927) - 4.45582 z (0.54 ک 6:574227 - 4.11818908 + 1-4183109) S (acosxda- -5.6030 integral normally, we get when we calculate

nay subintuval Divide [ls] into of length sn=1 with a = 1, 2, 3, 4, Sob following and points Now, fro) = fa)=f()) = cos1 = 4 flu) = 4 f2) = 8 cos2 = -3.32917 0.54030 2 f2) = 243) = 6 cas} =-5.93995 y f(3) = 444) = 16 cosy = -10-45029 Hay) - fb) = f15)-5coss = 1:4183009 Put these rales in - 1 3 0.54030 + (-3.3 2917) + (-5.93995) (-10-45829) +1.4183109 ] -5.9 2221 . Relative enor = 5.8927 +5.92229 -518927 -0.005 X lov Z. - - 0.5%