A Long, Thin Rod of Uniform Cross Section and Length

Multiple Choice

A long, thin rod of uniform cross section and length L has a density that depends on position along the bar. The linear density of the rod is given as A(1 - x/L) + B, where L is the distance from the left end of the rod. Determine the moment of inertia of the rod about an axis perpendicular to the rod that passes through the left end of the rod.

A) (A/4 + 2B/3) L²
B) (A/4 - B/3) L³
C) (-A/4 + B/3) L³
D) (-5A/4 + 2B/3) L³
E) (A/12 + B/3) L³