Question
Asked Dec 3, 2019

Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. Compare your result with the exact value of the integral. (Give your answers correct to 4 decimal places.)

 

253ln(x)dx; n=4

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Expert Answer

Step 1

Evaluate the definite integral using trapezoidal rule and Simpson’s rule. And compare with exact value.

Given:
5
J3ln (x)dx;n 4
Trapezoidal rule:
b
Дх
Хо
2
n-1
b-a
where, Ar
Simpson Rule:
b
Дк
)-re
[s(*)*+4s {y)+251x2) +4s(
f(x)dx
3
n-1
b-a
where, Ar
n
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Given: 5 J3ln (x)dx;n 4 Trapezoidal rule: b Дх Хо 2 n-1 b-a where, Ar Simpson Rule: b Дк )-re [s(*)*+4s {y)+251x2) +4s( f(x)dx 3 n-1 b-a where, Ar n

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Step 2

From the given information, a= 2, b= 5 and n=4.

b-а
п
5-2
Ar =
4
3
Ax = 0.75
4
End points:
-2,
=2+0.75 2.75
xz=2.75+0.75 =3.5
=3.5+0.75 4.25
-4.25+0.75 5
4
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b-а п 5-2 Ar = 4 3 Ax = 0.75 4 End points: -2, =2+0.75 2.75 xz=2.75+0.75 =3.5 =3.5+0.75 4.25 -4.25+0.75 5 4

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Step 3

Substitute known...

Trapezoidal rule:
5
In(x)dk =
2)-2s(%2)+25(ag)* S(%g)]
(2)+2 (2.75)+ 2f (3.5) +2f (4.25)+f (5)
2
31n(2)+2x3 n(2.75)+2x3 In(3.5)+2x3 In (4,25)+3 In(5)]
8
0.375 2.0794+6.0696+7.5166+8.6815+4.8283
10.9408
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Trapezoidal rule: 5 In(x)dk = 2)-2s(%2)+25(ag)* S(%g)] (2)+2 (2.75)+ 2f (3.5) +2f (4.25)+f (5) 2 31n(2)+2x3 n(2.75)+2x3 In(3.5)+2x3 In (4,25)+3 In(5)] 8 0.375 2.0794+6.0696+7.5166+8.6815+4.8283 10.9408

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Tagged in

Math

Calculus

Integration