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Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640

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BuyFindarrow_forward

Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640
Textbook Problem

In Exercises 1–14, use the Trapezoidal Rule and Simpson’s Rule to approximate the value of each definite integral. Compare your result with the exact value of the integral.

13. 1 2 ln x d x ; n = 4

To determine

To approximate: The value of definite integral by using Trapezoidal Rule and Simpson’s Rule and compare the result with the exact value of the integral.

Explanation

Given:

The integral 12lnxdx and n=4 .

Formula used:

Trapezoidal Rule: abf(x)dxΔx2[f(x0)+2f(x1)+...+2f(xn1)+f(xn)]

Where Δx=ban and xn=a+nΔx .

Simpson’s Rule: abf(x)dxΔx3[f(x)+4f(x1)+2f(x2)+...+4f(xn1)+f(xn)]

Where Δx=ban and n is even.

Calculation:

Consider the integral 12lnxdx .

Here, a=1, b=2 and n=4 .

Substitute a=1, b=2 and n=4 in Δx=ban ,

Δx=214=14

The end point of the sub-intervals are x0=1, x1=54, x2=32, x3=74 and x4=2 .

The Trapezoidal Rule yields,

12lnxdx142[f(1)+2f(54)+2f(32)+2f(74)+f(2)]=18[ln(1)+2(ln(54))+2(ln(32))+2(ln(74))+ln(2)]=18[0+(2ln(5)2ln(4))+(2ln(3)2ln(2))+(2ln(7)2ln(4))+ln(2)]=18[0+0.446287+0.810930+1.119232+0.693147]

Simplify further,

12lnxdx=3

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