#### To determine

**To find:**

The average density of the rod.

#### Answer

9 kg/m

#### 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-a∫abfx 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∫0812x+1 dx

=18∫0812x+1 dx

=128∫08x+1-12 dx

Integrating,

=128x+1121208

=3x+11208

=3912-0

=9 kg/m

**Conclusion:**

The average density of the rod 9 kg/m