082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

y
A
B
C
D
E
17.3 Simpson’s rule 501
quadraticcurveisfittedthroughthepointsC,DandE.Aftersomeanalysisanexpression
forapproximating the area isfound.
Simpson’s rulestates:
Figure17.2
InSimpson’s rulean
even number ofstrips
isused. Thecurve is
approximated by
quadratic curves.
x
area ≈ h 3 (y 0 +4y 1 +2y 2 +4y 3 +2y 4 +···+2y n−2 +4y n−1 +y n )
= h 3 {y 0 +4(y 1 +y 3 +···)+2(y 2 +y 4 +···)+y n }
wherenisan even number.
Example17.3 Estimate ∫ 1.3
0.5 e(x2) dx usingSimpson’srulewith (a)fourstrips, (b)eight strips.
Solution (a) We use Table 17.1 and notethath = 0.2.UsingSimpson’s rulewehave
estimatedvalue = 0.2
3 {1.2840+4(1.6323+3.3535)+2(2.2479)+5.4195}
= 2.0762
Using Simpson’s rulewe have found ∫ 1.3
0.5 e(x2) dx ≈ 2.0762.
(b) We use Table 17.2 and note thath = 0.1:
estimated value = 0.1 {1.2840 +4(1.4333 +1.8965 +2.7183 +4.2207)
3
+2(1.6323 +2.2479 +3.3535) +5.4195}
= 2.0749
Using Simpson’s rulewe have found ∫ 1.3
0.5 e(x2) dx ≈ 2.0749.
Example17.4 Estimate ∫ 2
1
√
1 +x3 dx using Simpson’s rulewith 10 strips.
Solution With10 stripsh = 0.1. UsingTable 17.4, wefind
estimatedvalue ≈ 0.1
3 {1.4142+4(1.5268+1.7880+2.0917+2.4317+2.8034)
+2(1.6517 +1.9349 +2.2574 +2.6138) +3.000}
= 2.130
In some cases the numerical values are not derived from a function but from actual
measurements.Numericalmethods can still beappliedinanidentical manner.