Question

Q7 Integral 15 Points Estimate the error from the trapezoidal rule and Simpson's rule when finding an approximation to Svada with 4 equally spaced subintervals of (0,1). Please select files) Select files)

Answer 1

1 Let T I = Isa da and f(x) = Voc. To calculate True value: :- formula fxd4 anti dx = ntl Irade - 's att de - 1½ tl a ½ti - 362 FU 2 [ph 3/2 -- 0 3 2 da. = 0.6667 ſeda True value of I is I tue - 0.6667

Numerical Integration Techniques :- a) Trapezoidal rule- Formula Hornh I fcandx - [4+%) +26+4+ h +/-) do n = number of intervale Not 4h - to da. do I fx) dx = 0.25 (145+ %)+2(3, +4,+%)] 25[(0+1)+7(0.5 +0.707+0.860) 0.25 2 T = 0.6432 I in Approximate value of I trapezoidal rule equals an I Trap = 0.6432

(b) Simpson's rule :- Formula :- Xothh 1 teelu -5 [4,444448 .**t they +2C%***+-- You) 7 do - - (tx) dx = (we de. 0 -0.25 3. $ [cb + )+4 (9,7%)+2() - 0.25 [60+ 1+ 4(0.540.86674-240-707) = 0.6565 : Approximate of I' in simpson's sule equals I simp=0.6565 as

To Estimate enors - (i) from Trapezoidal rule - Error in finding the value of geven definite integral in Trapezoidal rule is given by - I true I Trap = 0.6667 - 0.6432 E=0.0235 Eryoy from Trapezoidal rule Cii) From Simpson's yule :- Error from finding value of given definite integral in simpson's rule is given by Est I tme I simp -0.6667-0.6565 = 0.0102 Error from simpson's rule.