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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

A log 10 m long is cut at 1-meter intervals and its cross-sectional areas A (at a distance x from the end of the log) are listed in the table. Use the Midpoint Rule with n = 5 to estimate the volume of the log.

x ( m ) A ( m 2 )
0 0.68
1 0.65
2 0.64
3 0.61
4 0.58
5 0.59
6 0.53
7 0.55
8 0.52
9 0.50
10 0.48

To determine

To estimate:

The volume of the log by using the midpoint rule with  n=5

Explanation

1) Concept:

i. If the cross section is a disc and the radius of the disc is in terms of x  or y then area A=π radius2

ii. The volume of the solid revolution about the x-axis is

V= abA(x)dx

iii. Use the midpoint ruleto find the volume of the log.

Midpoint Rule:abf(x)dx=Mni=1nfxi-x=x[fxi-+ .+ fxn-]

where, x=b - an,n is the number of subintervals and  xi-=12xi-1+xi is midpoint of [xi-1, xi], xi=a+ix

2) Given:

A log 10 m long is cut at 1-meter intervals, and its cross-sectional areas are listed in the table below:

Also, n=5 is given

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