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

Review Exercises 17 505
EXERCISES17.3
1 Estimate the values ofthe following integrals using
Simpson’s rule:
∫ 3
(a) ln(x 3 +1)dx use10strips
2
∫ 2.6 √
(b) te t dt use eightstrips
1
2 Evaluate,usingthe trapeziumruleandSimpson’s
rule,
∫ 1
(a) (x 2 +1) 3/2 dx use four strips
0
∫ 1.6 sin2t
(b) dt use sixstrips
1 t
Solutions
1 (a) 2.7955 (b) 15.1164
2 (a) trapeziumrule: 1.5900,Simpson’s rule: 1.5681
(b) trapeziumrule:0.2464,Simpson’s rule:
0.2460
TechnicalComputingExercises17.3
1 (a) Plotthe fourthderivative of f (x) =
ln(x 3 +1)for2 x3.Useyourgraph
to findan upper bound for f (4) (x)for2x3.
Hence findan upperbound forthe error in
Question1(a) in Exercises17.3. Stateupper and
lower boundsforthe integral given in the
question.
(b) Repeat (a)with f (t) = √ te t for1 t 2.6.
Hence findan upper bound forthe error in Question1(b)
in Exercises17.3. Statelower and upper
boundsforthe integralgiven in the question.
2 Use MATLAB ® orasimilar technicalcomputing
language to findupper andlower boundsforthe
integrals in Question2in Exercises17.3.
REVIEWEXERCISES17
1 Iff(t)= √ t 2 +1find ∫ 2
1 f (t)dt using
(a) the trapezium rulewithh = 0.25
(b) Simpson’s ruleusingeightstrips.
2 Estimate the followingdefiniteintegrals usingthe
trapeziumrulewith sixstrips:
∫ 4 √
∫ 0.6
(a) x 3 +1dx (b) sin(t 2 )dt
1
0
∫ 0.8
∫ 0.3
(c) e (t2) 3
dt (d)
0.2
0 x 3 +2 dx
∫ 2.5 e t
(e) dt
1 t
3 Estimate the definiteintegrals in Question2using
Simpson’s rulewith sixstrips.
4 Estimate the followingdefiniteintegrals usingthe
trapeziumrule with eightstrips:
∫ 4
(a) cos( √ t)dt
0
∫ 6
(b) (t 2 +1) 3/2 dt
−2
∫ 3
(c) e √x dx
1
∫ 0.8
(d) tan(t 2 )dt
0
∫ 5
(e) ln(x 2 +1)dx
3