   Chapter 5.2, Problem 44E

Chapter
Section
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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