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Mathematical Applications for the Management, Life, and Social Sciences11th Edition

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Chapter 13.8, Problem 8E
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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Solutions

Chapter
Section
Problem 1CP
Problem 2CP
Problem 3CP
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
Problem 26E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
Problem 33E
Problem 34E
BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

For each integral in Problems 7-12, do the following.

(a) Approximate its value using the Trapezoidal Rule.

(b) Approximate its value using Simpson’s Rule.

(c) Find its exact value by integration.

(d) State which approximation is more accurate. (Round each result to 2 decimal places.)

∫ 1 0   x 3 d x ;    n = 4

(a)

To determine

To calculate: The approximate value of integral ∫01x3dx for n=4 by Trapezoidal Rule.

Explanation

Given information:

The provided integral ∫01x3dx for n=4.

Formula used:

The length of each n subdivision for a closed interval [a,b].

h=b−an

The approximate value of integral ∫abf(x)dx for n equal subdivision of closed interval [a,b] by Trapezoidal Rule is,

∫abf(x)dx≈h2[f(x0)+2f(x1)+2f(x2)+2f(x3)+⋯+f(xn)]

Where h is the length of each subdivision.

Calculation:

Consider the provided integral ∫01x3dx for n=4.

Since the number of equal subdivision is 4 and the lower and upper limit of the integral are 0 and 1 respectively.

n=4a=0b=1

Use the formula for the length of each n subdivision for a closed interval [a,b].

h=b−an

Substitute 0 for a, 1 for b and 4 for n.

h=1−04h=14

The corresponding value of xi=a+ih and f(xi)=xi2 for a=0 and h=14 is shown below

(b)

To determine

To calculate: The approximate value of integral ∫01x3dx for n=4 by Simpson Rule.

(c)

To determine

To calculate: The exact value of ∫01x3dx by integration.

(d)

To determine

The most accurate approximation rule among the Trapezoidal rule and Simpson Rule.

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