   Chapter 7.7, Problem 28E

Chapter
Section
Textbook Problem

# Find the approximations T n , M n , and S n for n = 6 and 12. Then compute the corresponding errors E T , E M , and E S . (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations can you make? In particular, what happens to the errors when n is doubled? ∫ 1 4 1 x   d x To determine

To evaluate: To find the approximation Tn,Mn,Sn for n=6, n=12 and compute the corresponding errors ET,EM,ES for the integral 141xdx

Explanation

Formula used:

Tn=i=0n1Δ2[f(a+iΔ)+(a+(i+1))Δ]Mn=i=0n1Δf(a+(i+12)Δ)Sn=23Mn+13Tn

b, a are limits of the given integral

Kindly note that the Simpson’s formula Sn given in the text book can be simplified to Sn=23Mn+13Tn

Calculations:

Given integral 141xdx

Exact value of the integral is

(2x)14=2

(a) For n=6

We have b=4, a=1, n=6  so Δ=12

Tn=i=05Δ2[f(a+iΔ)+f(a+(i+1))Δ]=14[f(1)+2f(32)+2f(42)+2f(52)+2f(62)+2f(72)+f(82)]=14[1+2×23+2×24+2×25+2×26+2×27+28]

Tn=12[13+14+15+16+17]+14[1+14]=2.008965

Mn=i=05Δ[f(a+(i+12)Δ)]=12[f(54)+f(74)+f(94)++f(154)]=12[45+47+49++415]=[15+17+19+111+113+115]=1.99557

Sn=23Mn+13Tn=2.000035

ET=22.008965=.0089649EM=21.99557=.004429ES=22.000035=

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