   Chapter 5.5, Problem 19E

Chapter
Section
Textbook Problem

# The linear density in a rod 8 m long is 12 / x + 1  kg/m , where x is measured in meters from one end of the rod. Find the average density of the rod.

To determine

To find:

The average density of the rod.

Explanation

1) Concept:

Use the mean value theorem for integral to find the average density of the rod.

2) Mean Value Theorem for Integrals

If f is continuous on a,b then there exists a number c in a,b such that

fc=fave=1b-aabfx dx

that is,abfx dx=fcb-a

3) Given:

i) The linear density in a rod is fx=12x+1 kg/m

ii) The length of the rod is 8m

4) Calculation:

The total length of the rod is 8m.

So, here,a=0m and b=8m

Average density of the rod

=18-0

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