Question

Estimate the error from the trapezoidal rule and Simpson's rule when finding an approximation to So Vædæ with 4 equally spaced subintervals of (0,1).

Answer 1

Solna Formula for 6770 in Trapezoida Rule for integral. b Sb f(x) dx Where You yo Yn are the values of y = f(x) at No, a an. Where 20 = Kotnh ha boa n. is given by error for interval (xo, x) 12 error for interval (x 2 ) is = -h3 you -h3 y." th? Y"n=1 ) 12 12 + Here. The 1-0 4 error in (an-1, and is = total error is given by Ex-h3 (Yory" + Y'n=1) y f(x)=√x. 10 x = 0.25 l = 0.5 x = 0.75 x4 y' (x) érz and y! (x)=ļx- 0.25 Xo = 0 I. Now U こ 1 x-3. (2

=0 y" (X.) = y"(0) y" (91) y" (0.25) = -2. y" (0.25) = - 1 / 1 Х (0.25)-32 - 1 (0.5) -372 4 yul0.5) y" (X) = "10.5) 3 =-0.7071 y" (x3) - y"(0.75) = (0.75 8960-75) = -0.3849 y"(04-) = y" (1) -0.25 11 x - 3/2 4 - -1 4 (1)-312 y" :) Then. Error. is given by -h3 s Ez 12 Y"co) + y "(0.25)+ y"los) + y "(0-75) + y" (1) (0.253 = 0-2 -0.7071 12 -0.3849 -0.25 (0.2513 x 3.342 12

ES 0.00 43515625 Anst Hence, Error in integral Calculation Sa dx by Trapezoidal sule for 4 Sub interval is EN0.004351 Simpson's rule 54 go error go error go Nowe Error formula of integral by is given by principal part of error in [no, xz] is = the hyo in [22, xq] is =-45 yt e Total in (xo, xq] is - hs sy ty t } Differentiating (2) writx, we have -512 (3) Again differentiating (3) wort & gt (x) = 2 x - x - x - 7/2 (4) yt (x) = y(0) yh(x) = 4+ (0.25) = - 15 (0.25) The 1 y"? () X-3 x 2 X-5 x 2 - y to yt 16

Date Page - 15 x 128. 16 y (0.25) = - 120 Error E= -hs ( y (0) + y + (0.25] go (0.25.354 120 20) go. (0.255 x 120 E. - 0.001302083 Auss Idence, Error from Simpson's rule of approximating integral forn dx is given by E 0.001302