A two part question on simpsons rule. Please explain your answer part 1 For the

Question

A two part question on simpsons rule. Please explain your answer

part 1 For the function ?_0^2??f(x)=(x-2)^2+2 ? dx we are going to estimate the area under the curve using the Simpson%u2019s rule where n = 6 by doing the following:

?x= (b-a)/n =

x_1=

x_2=

x_3=

x_4=

x_5=

x_6=

x_7=

f(x_1 )=

f(x_2 )=

f(x_3 )=

f(x_4 )=

f(x_5 )=

f(x_6 )=

f(x_7 )=

The formula for Simpson%u2019s Rule is ?_a^b???x/3 (f(x_1 )+4f(x_2 )+2f(x_3 )+4f(x_3 )+2f(x_4 )+?+4f(x_(n-1) )+f(x_n)) ?. Plug the values above into Simpson%u2019s Rule and calculate your approximation.

part 2 Which formula, Trapezoidal or Simpson%u2019s, gives the better approximation?

Explanation / Answer

y = ln(3+x^3)
dy/dx = 3x^2/(3+x^3)
(dy/dx)^2 = 9x^4 /(3+x^3)^2

Arc length = sqrt [ 1+(dy/dx)^2] = sqrt (1 + 9x^4 /(3+x^3)^2 )
We want to integrate this from 0 to 5 for which we need to use Simpson's Rule with n=10

Lower limit 0
Upper limit 5
Number of intervals 10
%u0394x = (5 - 0) / 10 = 0.5
number of intervals 10
The intervals are:

(0,0.5), (0.5,1), (1,1.5), (1.5,2),
(2,2.5), (2.5,3), (3,3.5), (3.5,4),
(4,4.5), (4.5,5),

The midpoints of each interval are
0.25, 0.75, 1.25, 1.75,
2.25, 2.75, 3.25, 3.75, 4.25,
4.75,

Simpson's Sum =
f(0) +2f(0.5) + 2f(1) + 2f(1.5) + 2f(2) + 2f(2.5) +
2f(3) + 2f(3.5) + 2f(4) + 2f(4.5) + f(5) +

4f(0.25) + 4f(0.75) + 4f(1.25) + 4f(1.75) + 4f(2.25) +
4f(2.75) + 4f(3.25) + 4f(3.75) + 4f(4.25) + f(4.75)


Simpson's Sum =
1 + 2(1.028397) + 2(1.25) + 2(1.456402) + 2(1.479893) + 2(1.418967) +
2(1.345362) + 2(1.281306) + 2(1.230144) + 2(1.190195) + 1.159018 +

4(1.001931) + 4(1.114988) + 4(1.376815) + 4(1.485915) + 4(1.453897) +
4(1.381643) + 4(1.311722) + 4(1.254175) + 4(1.208923) + 4(1.173654)


Simpson's sum = %u0394x/6 [f(a[0])+2%u03A3 sum of end points+4%u03A3 sum of midpoints+f(a[n])]
=(0.5/6)[1 +25.520352 +51.054655 +1.159018]
= (0.083333)(76.575006) = 6.38125052
The arc length using Simpson's rule is 6.3813

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