what is the effective coefficient of expansion of a homogenous rod whose coefficient varies from alpha 1 to alpha 2 with length.

use integration for the fact

let length of rod be l

assuming our point of consideration to be A having a1 

the take a element dx at a distance x from A

variation of alpha from A is given by:-

instantaneous alpha   =  a1  + (a2-a1)x/L

now alpha effective = ∫(inst.alpha * dx) / L

solve to get the answer

i leave it to you...