The coefficient of linear expansion of homogeneous rod changes linearly from α1 to α2 from one end to the other end. The effective coefficient of linear expansion is

At a distance x,  αx=α1+α2-α1lx total expansion of the rod for a rise in temperature Δθ.   ∆l=∫0ldxα1+α2-α1lx∆θ       =α1+α22l∆θ It should be noted that, if a quantity changes line only from a minimum value to a maximum value, then its average value is the arithmetic mean of the two.