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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it.

A 10-ft chain weighs 25 lb and hangs from a ceiling. Find the work done in lifting the lower end of the chain to the ceiling so that it is level with the upper end.

To determine

To find:

Work done in lifting the lower end of the chain to the ceiling

Explanation

1) Concept:

Approximate the required work by using the concept of Riemann sum. Then express the work as an integral and evaluate it.

2) Given:

i) Weight of the chain 25 lb

ii) Length of the chain 10 ft

3) Calculation:

Let’s place the origin at the top of the ceiling and x- axis pointing downward as in the figure.Divide the cable into small parts with length x

If xi* is a point in the ith such interval, then all points in the interval are lifted by approximately the same amount, xi*

Total Length of chain 10 ft and total weight of chain 25 lb

The length of chain per foot is 25lb/10ft=2.5 lb/ft

The weight of the ith part is (2.5 lb/ft)(xft)=2.5 x lb

Divide the length of chain 10 ft into equal parts 0510

The upper part of the chain is 0xi*5 and lower part is 5xi*10

To find the work done in lifting the lower end of the chain to the ceiling so that it is level with the upper end.

The lower part of the chain is 5xi*10

Distance of point xi* in the interval 5xi*10 is xi*-5ft

When lifting the point xi* will travel 2(xi*-5) ft

Therefore,

The lower part of the chain ( for 5xi*10 ) has to be lifted 2(xi*-5) ft,

The work needed to lift for the ith subinterval of the chain is 2(xi*-5)(2

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