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CalculusCalculus (MindTap Course List)8th Edition

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Chapter 4.2, Problem 31E
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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Solutions

Chapter
Section
Problem 1E
Problem 2E
Problem 3E
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Problem 5E
Problem 6E
Problem 7E
Problem 8E
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Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
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Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
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Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
Problem 33E
Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
Problem 39E
Problem 40E
Problem 41E
Problem 42E
Problem 43E
Problem 44E
Problem 45E
Problem 46E
Problem 47E
Problem 48E
Problem 49E
Problem 50E
Problem 51E
Problem 52E
Problem 53E
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Problem 55E
Problem 56E
Problem 57E
Problem 58E
Problem 59E
Problem 60E
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Problem 62E
Problem 63E
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Problem 71E
Problem 72E
Problem 73E
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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Express the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit.

∫ 0 π sin 5 x   d x

To determine

To express:

The integral ∫0πsin⁡5x dx as a limit of the sums then evaluate by using a computer’s algebra system to find both the sum and the limit.

Explanation

1) Concept:

Use theorem (4) to express the given integral as a limit of the sums then use a computer’s algebra system to find both the sum and the limit.

Theorem (4):

If f is integrable on [a, b] then

∫abfxdx=limn→∞⁡∑i=1nfxi ∆x

Where ∆x= b - an and xi=a+i ∆x

Also, ∑i=1nf(xi*)∆x is the Riemann sum formula.

2) Given:

∫0πsin⁡5x dx

3) Calculation:

Comparing the given integral with theorem (4), set

a=0, b=π and fx=sin⁡5x

Substituting the values of a and b in ∆x,

∆x= b-an

∆x= πn

Now, find xi.

xi=a+i ∆x

 xi=πin

By using theorem (4) and substituting the values of ∆x and xi, fxi=sin⁡5xi=sin⁡5

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