Question

4. -1 POINIS Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n Round your answer to four decimal places and compare the results with the exact value of the definite integral dx, 4 Trapezoidal Simpson's exact Need Help? Read Talkie Tur

Answer 1

3 fa o Sfus dx ~ At u 4 3 xa ²+1 dx ,n=4 Trapezoidal rule states that bi 142 (f(x0) +2 +2 f(x2) + +--- ef (Any)+ flaw where anabra ta=0, 623, n=4 Ax= 3-0 = 3 Divide the integral [0,3 into nzu subintervals of length An= 3 with the endpoints 3 3=b f(10) = f(a)= f(0) = 0 24(2,3=24 (6)=zCaver IS 2 f(*2)=24(*)-2[zalni ef(43) = 2445)=2(4,142+1}=2997 f(uu)= $(6)= $(3)= 3v1+1 = 3 STO xx+idx= 314 lot & + 30+ 4งจร + 3STO) -(0.44379 प 9 azo, ū 1 z 1 ho ū 313 - 2 IS 8

Simpson's Rule a 3 ob f(x)dx AX (f(nolt hef(m) + 24(12)thfcxz) + 2 f/nn-2)+4 Hunt) + f(n)), Dx=b-a we have Then Ax= 3 and endpoints a=o, b=3 h=4 n 3-0 ml are same as trapezoidal Rule fluo)=f(a) = f(0) = 0 ufem.) zu f(3) = u li j(212 +13 -15 is u -3113 2 ajat 2 f (12)=24E)=2(NE) uf (Mg) = f (*)=4[v@rize}:44 (nu)= f(b) = f(3) = 3010 finally just sumup the above and multiply by an (ot 997 touto 3 ų TS 3513 + ť +8570) 2 =10.2012 72u a

Now calculate the exact value f arx²t da Substitute u=x2+1 then du =22 ) dx = ex du an unt = Sxra²+1 dx = 2 Sru du apply power rule fundua nti 늙 Undo substitution ce=x²+1 Vatt 312 ) = lú 3 - 1/2+1 (x²+1)/2 312 3 0 ta li 3 arazti dna (12) ${63)+1) 92 - (03+1 a [ 1032 ] 10. 2075922 Trapezoidal Simpson's 10.2013 Exact [10.2076 10:4438